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Theorem domss2 9138
Description: A corollary of disjenex 9137. If 𝐹 is an injection from 𝐴 to 𝐡 then 𝐺 is a right inverse of 𝐹 from 𝐡 to a superset of 𝐴. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
Hypothesis
Ref Expression
domss2.1 𝐺 = β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})))
Assertion
Ref Expression
domss2 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐺:𝐡–1-1-ontoβ†’ran 𝐺 ∧ 𝐴 βŠ† ran 𝐺 ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ 𝐴)))
Proof of Theorem domss2
StepHypRef Expression
1 f1f1orn 6843 . . . . . . . 8 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)
213ad2ant1 1131 . . . . . . 7 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)
3 simp2 1135 . . . . . . . . . 10 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ 𝐴 ∈ 𝑉)
4 rnexg 7897 . . . . . . . . . 10 (𝐴 ∈ 𝑉 β†’ ran 𝐴 ∈ V)
53, 4syl 17 . . . . . . . . 9 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ ran 𝐴 ∈ V)
6 uniexg 7732 . . . . . . . . 9 (ran 𝐴 ∈ V β†’ βˆͺ ran 𝐴 ∈ V)
7 pwexg 5375 . . . . . . . . 9 (βˆͺ ran 𝐴 ∈ V β†’ 𝒫 βˆͺ ran 𝐴 ∈ V)
85, 6, 73syl 18 . . . . . . . 8 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ 𝒫 βˆͺ ran 𝐴 ∈ V)
9 1stconst 8088 . . . . . . . 8 (𝒫 βˆͺ ran 𝐴 ∈ V β†’ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})):((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})–1-1-ontoβ†’(𝐡 βˆ– ran 𝐹))
108, 9syl 17 . . . . . . 7 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})):((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})–1-1-ontoβ†’(𝐡 βˆ– ran 𝐹))
11 difexg 5326 . . . . . . . . . 10 (𝐡 ∈ π‘Š β†’ (𝐡 βˆ– ran 𝐹) ∈ V)
12113ad2ant3 1133 . . . . . . . . 9 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐡 βˆ– ran 𝐹) ∈ V)
13 disjen 9136 . . . . . . . . 9 ((𝐴 ∈ 𝑉 ∧ (𝐡 βˆ– ran 𝐹) ∈ V) β†’ ((𝐴 ∩ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) = βˆ… ∧ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}) β‰ˆ (𝐡 βˆ– ran 𝐹)))
143, 12, 13syl2anc 582 . . . . . . . 8 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ ((𝐴 ∩ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) = βˆ… ∧ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}) β‰ˆ (𝐡 βˆ– ran 𝐹)))
1514simpld 493 . . . . . . 7 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐴 ∩ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) = βˆ…)
16 disjdif 4470 . . . . . . . 8 (ran 𝐹 ∩ (𝐡 βˆ– ran 𝐹)) = βˆ…
1716a1i 11 . . . . . . 7 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (ran 𝐹 ∩ (𝐡 βˆ– ran 𝐹)) = βˆ…)
18 f1oun 6851 . . . . . . 7 (((𝐹:𝐴–1-1-ontoβ†’ran 𝐹 ∧ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})):((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})–1-1-ontoβ†’(𝐡 βˆ– ran 𝐹)) ∧ ((𝐴 ∩ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) = βˆ… ∧ (ran 𝐹 ∩ (𝐡 βˆ– ran 𝐹)) = βˆ…)) β†’ (𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))–1-1-ontoβ†’(ran 𝐹 βˆͺ (𝐡 βˆ– ran 𝐹)))
192, 10, 15, 17, 18syl22anc 835 . . . . . 6 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))–1-1-ontoβ†’(ran 𝐹 βˆͺ (𝐡 βˆ– ran 𝐹)))
20 undif2 4475 . . . . . . . 8 (ran 𝐹 βˆͺ (𝐡 βˆ– ran 𝐹)) = (ran 𝐹 βˆͺ 𝐡)
21 f1f 6786 . . . . . . . . . . 11 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴⟢𝐡)
22213ad2ant1 1131 . . . . . . . . . 10 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ 𝐹:𝐴⟢𝐡)
2322frnd 6724 . . . . . . . . 9 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ ran 𝐹 βŠ† 𝐡)
24 ssequn1 4179 . . . . . . . . 9 (ran 𝐹 βŠ† 𝐡 ↔ (ran 𝐹 βˆͺ 𝐡) = 𝐡)
2523, 24sylib 217 . . . . . . . 8 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (ran 𝐹 βˆͺ 𝐡) = 𝐡)
2620, 25eqtrid 2782 . . . . . . 7 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (ran 𝐹 βˆͺ (𝐡 βˆ– ran 𝐹)) = 𝐡)
2726f1oeq3d 6829 . . . . . 6 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ ((𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))–1-1-ontoβ†’(ran 𝐹 βˆͺ (𝐡 βˆ– ran 𝐹)) ↔ (𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))–1-1-onto→𝐡))
2819, 27mpbid 231 . . . . 5 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))–1-1-onto→𝐡)
29 f1ocnv 6844 . . . . 5 ((𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))–1-1-onto→𝐡 β†’ β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1-ontoβ†’(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})))
3028, 29syl 17 . . . 4 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1-ontoβ†’(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})))
31 domss2.1 . . . . 5 𝐺 = β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})))
32 f1oeq1 6820 . . . . 5 (𝐺 = β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) β†’ (𝐺:𝐡–1-1-ontoβ†’(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) ↔ β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1-ontoβ†’(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))))
3331, 32ax-mp 5 . . . 4 (𝐺:𝐡–1-1-ontoβ†’(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) ↔ β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1-ontoβ†’(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})))
3430, 33sylibr 233 . . 3 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ 𝐺:𝐡–1-1-ontoβ†’(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})))
35 f1ofo 6839 . . . . 5 (𝐺:𝐡–1-1-ontoβ†’(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†’ 𝐺:𝐡–ontoβ†’(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})))
36 forn 6807 . . . . 5 (𝐺:𝐡–ontoβ†’(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†’ ran 𝐺 = (𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})))
3734, 35, 363syl 18 . . . 4 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ ran 𝐺 = (𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})))
3837f1oeq3d 6829 . . 3 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐺:𝐡–1-1-ontoβ†’ran 𝐺 ↔ 𝐺:𝐡–1-1-ontoβ†’(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))))
3934, 38mpbird 256 . 2 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ 𝐺:𝐡–1-1-ontoβ†’ran 𝐺)
40 ssun1 4171 . . 3 𝐴 βŠ† (𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))
4140, 37sseqtrrid 4034 . 2 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ 𝐴 βŠ† ran 𝐺)
42 ssid 4003 . . . 4 ran 𝐹 βŠ† ran 𝐹
43 cores 6247 . . . 4 (ran 𝐹 βŠ† ran 𝐹 β†’ ((𝐺 β†Ύ ran 𝐹) ∘ 𝐹) = (𝐺 ∘ 𝐹))
4442, 43ax-mp 5 . . 3 ((𝐺 β†Ύ ran 𝐹) ∘ 𝐹) = (𝐺 ∘ 𝐹)
45 dmres 6002 . . . . . . . . 9 dom (β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹) = (ran 𝐹 ∩ dom β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})))
46 f1ocnv 6844 . . . . . . . . . . . 12 ((1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})):((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})–1-1-ontoβ†’(𝐡 βˆ– ran 𝐹) β†’ β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})):(𝐡 βˆ– ran 𝐹)–1-1-ontoβ†’((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))
47 f1odm 6836 . . . . . . . . . . . 12 (β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})):(𝐡 βˆ– ran 𝐹)–1-1-ontoβ†’((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}) β†’ dom β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) = (𝐡 βˆ– ran 𝐹))
4810, 46, 473syl 18 . . . . . . . . . . 11 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ dom β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) = (𝐡 βˆ– ran 𝐹))
4948ineq2d 4211 . . . . . . . . . 10 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (ran 𝐹 ∩ dom β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) = (ran 𝐹 ∩ (𝐡 βˆ– ran 𝐹)))
5049, 16eqtrdi 2786 . . . . . . . . 9 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (ran 𝐹 ∩ dom β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) = βˆ…)
5145, 50eqtrid 2782 . . . . . . . 8 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ dom (β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹) = βˆ…)
52 relres 6009 . . . . . . . . 9 Rel (β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹)
53 reldm0 5926 . . . . . . . . 9 (Rel (β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹) β†’ ((β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹) = βˆ… ↔ dom (β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹) = βˆ…))
5452, 53ax-mp 5 . . . . . . . 8 ((β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹) = βˆ… ↔ dom (β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹) = βˆ…)
5551, 54sylibr 233 . . . . . . 7 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹) = βˆ…)
5655uneq2d 4162 . . . . . 6 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (◑𝐹 βˆͺ (β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹)) = (◑𝐹 βˆͺ βˆ…))
57 cnvun 6141 . . . . . . . . 9 β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) = (◑𝐹 βˆͺ β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})))
5831, 57eqtri 2758 . . . . . . . 8 𝐺 = (◑𝐹 βˆͺ β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})))
5958reseq1i 5976 . . . . . . 7 (𝐺 β†Ύ ran 𝐹) = ((◑𝐹 βˆͺ β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) β†Ύ ran 𝐹)
60 resundir 5995 . . . . . . 7 ((◑𝐹 βˆͺ β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) β†Ύ ran 𝐹) = ((◑𝐹 β†Ύ ran 𝐹) βˆͺ (β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹))
61 df-rn 5686 . . . . . . . . . 10 ran 𝐹 = dom ◑𝐹
6261reseq2i 5977 . . . . . . . . 9 (◑𝐹 β†Ύ ran 𝐹) = (◑𝐹 β†Ύ dom ◑𝐹)
63 relcnv 6102 . . . . . . . . . 10 Rel ◑𝐹
64 resdm 6025 . . . . . . . . . 10 (Rel ◑𝐹 β†’ (◑𝐹 β†Ύ dom ◑𝐹) = ◑𝐹)
6563, 64ax-mp 5 . . . . . . . . 9 (◑𝐹 β†Ύ dom ◑𝐹) = ◑𝐹
6662, 65eqtri 2758 . . . . . . . 8 (◑𝐹 β†Ύ ran 𝐹) = ◑𝐹
6766uneq1i 4158 . . . . . . 7 ((◑𝐹 β†Ύ ran 𝐹) βˆͺ (β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹)) = (◑𝐹 βˆͺ (β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹))
6859, 60, 673eqtrri 2763 . . . . . 6 (◑𝐹 βˆͺ (β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹)) = (𝐺 β†Ύ ran 𝐹)
69 un0 4389 . . . . . 6 (◑𝐹 βˆͺ βˆ…) = ◑𝐹
7056, 68, 693eqtr3g 2793 . . . . 5 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐺 β†Ύ ran 𝐹) = ◑𝐹)
7170coeq1d 5860 . . . 4 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ ((𝐺 β†Ύ ran 𝐹) ∘ 𝐹) = (◑𝐹 ∘ 𝐹))
72 f1cocnv1 6862 . . . . 5 (𝐹:𝐴–1-1→𝐡 β†’ (◑𝐹 ∘ 𝐹) = ( I β†Ύ 𝐴))
73723ad2ant1 1131 . . . 4 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (◑𝐹 ∘ 𝐹) = ( I β†Ύ 𝐴))
7471, 73eqtrd 2770 . . 3 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ ((𝐺 β†Ύ ran 𝐹) ∘ 𝐹) = ( I β†Ύ 𝐴))
7544, 74eqtr3id 2784 . 2 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐺 ∘ 𝐹) = ( I β†Ύ 𝐴))
7639, 41, 753jca 1126 1 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐺:𝐡–1-1-ontoβ†’ran 𝐺 ∧ 𝐴 βŠ† ran 𝐺 ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  Vcvv 3472   βˆ– cdif 3944   βˆͺ cun 3945   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321  π’« cpw 4601  {csn 4627  βˆͺ cuni 4907   class class class wbr 5147   I cid 5572   Γ— cxp 5673  β—‘ccnv 5674  dom cdm 5675  ran crn 5676   β†Ύ cres 5677   ∘ ccom 5679  Rel wrel 5680  βŸΆwf 6538  β€“1-1β†’wf1 6539  β€“ontoβ†’wfo 6540  β€“1-1-ontoβ†’wf1o 6541  1st c1st 7975   β‰ˆ cen 8938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-1st 7977  df-2nd 7978  df-en 8942
This theorem is referenced by:  domssex2  9139  domssex  9140
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