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Theorem 3f1oss1 46975
Description: The composition of three bijections as bijection from the image of the domain onto the image of the range of the middle bijection. (Contributed by AV, 15-Aug-2025.)
Assertion
Ref Expression
3f1oss1 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → ((𝐻𝐺) ∘ 𝐹):(𝐹𝐶)–1-1-onto→(𝐻𝐷))

Proof of Theorem 3f1oss1
StepHypRef Expression
1 f1ocnv 6855 . . . . . . . 8 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
2 f1of1 6842 . . . . . . . 8 (𝐹:𝐵1-1-onto𝐴𝐹:𝐵1-1𝐴)
31, 2syl 17 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1𝐴)
433ad2ant1 1131 . . . . . 6 ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) → 𝐹:𝐵1-1𝐴)
54adantr 480 . . . . 5 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → 𝐹:𝐵1-1𝐴)
6 cnvimass 6096 . . . . . . . 8 (𝐹𝐶) ⊆ dom 𝐹
7 f1of 6843 . . . . . . . . 9 (𝐹:𝐵1-1-onto𝐴𝐹:𝐵𝐴)
8 fdm 6740 . . . . . . . . . 10 (𝐹:𝐵𝐴 → dom 𝐹 = 𝐵)
98eqcomd 2739 . . . . . . . . 9 (𝐹:𝐵𝐴𝐵 = dom 𝐹)
101, 7, 93syl 18 . . . . . . . 8 (𝐹:𝐴1-1-onto𝐵𝐵 = dom 𝐹)
116, 10sseqtrrid 4049 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵 → (𝐹𝐶) ⊆ 𝐵)
12113ad2ant1 1131 . . . . . 6 ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) → (𝐹𝐶) ⊆ 𝐵)
1312adantr 480 . . . . 5 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → (𝐹𝐶) ⊆ 𝐵)
14 f1ofn 6844 . . . . . . . . . 10 (𝐹:𝐵1-1-onto𝐴𝐹 Fn 𝐵)
151, 14syl 17 . . . . . . . . 9 (𝐹:𝐴1-1-onto𝐵𝐹 Fn 𝐵)
16153ad2ant1 1131 . . . . . . . 8 ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) → 𝐹 Fn 𝐵)
1716adantr 480 . . . . . . 7 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → 𝐹 Fn 𝐵)
18 eqidd 2734 . . . . . . 7 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → (ran 𝐹𝐶) = (ran 𝐹𝐶))
19 eqidd 2734 . . . . . . 7 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → (𝐹𝐶) = (𝐹𝐶))
2017, 18, 19rescnvimafod 7087 . . . . . 6 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → (𝐹 ↾ (𝐹𝐶)):(𝐹𝐶)–onto→(ran 𝐹𝐶))
21 fof 6815 . . . . . 6 ((𝐹 ↾ (𝐹𝐶)):(𝐹𝐶)–onto→(ran 𝐹𝐶) → (𝐹 ↾ (𝐹𝐶)):(𝐹𝐶)⟶(ran 𝐹𝐶))
2220, 21syl 17 . . . . 5 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → (𝐹 ↾ (𝐹𝐶)):(𝐹𝐶)⟶(ran 𝐹𝐶))
23 f1resf1 6807 . . . . 5 ((𝐹:𝐵1-1𝐴 ∧ (𝐹𝐶) ⊆ 𝐵 ∧ (𝐹 ↾ (𝐹𝐶)):(𝐹𝐶)⟶(ran 𝐹𝐶)) → (𝐹 ↾ (𝐹𝐶)):(𝐹𝐶)–1-1→(ran 𝐹𝐶))
245, 13, 22, 23syl3anc 1369 . . . 4 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → (𝐹 ↾ (𝐹𝐶)):(𝐹𝐶)–1-1→(ran 𝐹𝐶))
25 f1of1 6842 . . . . . . . 8 (𝐺:𝐶1-1-onto𝐷𝐺:𝐶1-1𝐷)
26253ad2ant2 1132 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) → 𝐺:𝐶1-1𝐷)
2726adantr 480 . . . . . 6 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → 𝐺:𝐶1-1𝐷)
28 inss2 4246 . . . . . 6 (ran 𝐹𝐶) ⊆ 𝐶
29 f1ores 6857 . . . . . 6 ((𝐺:𝐶1-1𝐷 ∧ (ran 𝐹𝐶) ⊆ 𝐶) → (𝐺 ↾ (ran 𝐹𝐶)):(ran 𝐹𝐶)–1-1-onto→(𝐺 “ (ran 𝐹𝐶)))
3027, 28, 29sylancl 585 . . . . 5 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → (𝐺 ↾ (ran 𝐹𝐶)):(ran 𝐹𝐶)–1-1-onto→(𝐺 “ (ran 𝐹𝐶)))
31 f1ofo 6850 . . . . . . . . . . . . . 14 (𝐹:𝐵1-1-onto𝐴𝐹:𝐵onto𝐴)
32 forn 6818 . . . . . . . . . . . . . 14 (𝐹:𝐵onto𝐴 → ran 𝐹 = 𝐴)
331, 31, 323syl 18 . . . . . . . . . . . . 13 (𝐹:𝐴1-1-onto𝐵 → ran 𝐹 = 𝐴)
34333ad2ant1 1131 . . . . . . . . . . . 12 ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) → ran 𝐹 = 𝐴)
3534adantr 480 . . . . . . . . . . 11 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → ran 𝐹 = 𝐴)
3635ineq1d 4227 . . . . . . . . . 10 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → (ran 𝐹𝐶) = (𝐴𝐶))
37 incom 4217 . . . . . . . . . . . 12 (𝐴𝐶) = (𝐶𝐴)
38 dfss2 3981 . . . . . . . . . . . . 13 (𝐶𝐴 ↔ (𝐶𝐴) = 𝐶)
3938biimpi 216 . . . . . . . . . . . 12 (𝐶𝐴 → (𝐶𝐴) = 𝐶)
4037, 39eqtrid 2785 . . . . . . . . . . 11 (𝐶𝐴 → (𝐴𝐶) = 𝐶)
4140ad2antrl 727 . . . . . . . . . 10 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → (𝐴𝐶) = 𝐶)
4236, 41eqtrd 2773 . . . . . . . . 9 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → (ran 𝐹𝐶) = 𝐶)
4342imaeq2d 6074 . . . . . . . 8 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → (𝐺 “ (ran 𝐹𝐶)) = (𝐺𝐶))
44 f1ofn 6844 . . . . . . . . . . . 12 (𝐺:𝐶1-1-onto𝐷𝐺 Fn 𝐶)
45 fnima 6694 . . . . . . . . . . . 12 (𝐺 Fn 𝐶 → (𝐺𝐶) = ran 𝐺)
4644, 45syl 17 . . . . . . . . . . 11 (𝐺:𝐶1-1-onto𝐷 → (𝐺𝐶) = ran 𝐺)
47 f1ofo 6850 . . . . . . . . . . . 12 (𝐺:𝐶1-1-onto𝐷𝐺:𝐶onto𝐷)
48 forn 6818 . . . . . . . . . . . 12 (𝐺:𝐶onto𝐷 → ran 𝐺 = 𝐷)
4947, 48syl 17 . . . . . . . . . . 11 (𝐺:𝐶1-1-onto𝐷 → ran 𝐺 = 𝐷)
5046, 49eqtrd 2773 . . . . . . . . . 10 (𝐺:𝐶1-1-onto𝐷 → (𝐺𝐶) = 𝐷)
51503ad2ant2 1132 . . . . . . . . 9 ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) → (𝐺𝐶) = 𝐷)
5251adantr 480 . . . . . . . 8 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → (𝐺𝐶) = 𝐷)
5343, 52eqtrd 2773 . . . . . . 7 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → (𝐺 “ (ran 𝐹𝐶)) = 𝐷)
5453eqcomd 2739 . . . . . 6 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → 𝐷 = (𝐺 “ (ran 𝐹𝐶)))
5554f1oeq3d 6840 . . . . 5 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → ((𝐺 ↾ (ran 𝐹𝐶)):(ran 𝐹𝐶)–1-1-onto𝐷 ↔ (𝐺 ↾ (ran 𝐹𝐶)):(ran 𝐹𝐶)–1-1-onto→(𝐺 “ (ran 𝐹𝐶))))
5630, 55mpbird 257 . . . 4 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → (𝐺 ↾ (ran 𝐹𝐶)):(ran 𝐹𝐶)–1-1-onto𝐷)
57 f1orel 6846 . . . . . . . . . . 11 (𝐹:𝐴1-1-onto𝐵 → Rel 𝐹)
58573ad2ant1 1131 . . . . . . . . . 10 ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) → Rel 𝐹)
5958adantr 480 . . . . . . . . 9 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → Rel 𝐹)
60 dfrel2 6205 . . . . . . . . 9 (Rel 𝐹𝐹 = 𝐹)
6159, 60sylib 218 . . . . . . . 8 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → 𝐹 = 𝐹)
6261eqcomd 2739 . . . . . . 7 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → 𝐹 = 𝐹)
6362imaeq1d 6073 . . . . . 6 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → (𝐹𝐶) = (𝐹𝐶))
6463f1oeq2d 6839 . . . . 5 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → ((𝐺𝐹):(𝐹𝐶)–1-1-onto𝐷 ↔ (𝐺𝐹):(𝐹𝐶)–1-1-onto𝐷))
651, 7syl 17 . . . . . . . 8 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵𝐴)
66653ad2ant1 1131 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) → 𝐹:𝐵𝐴)
6766adantr 480 . . . . . 6 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → 𝐹:𝐵𝐴)
68 eqid 2733 . . . . . 6 (ran 𝐹𝐶) = (ran 𝐹𝐶)
69 eqid 2733 . . . . . 6 (𝐹𝐶) = (𝐹𝐶)
70 eqid 2733 . . . . . 6 (𝐹 ↾ (𝐹𝐶)) = (𝐹 ↾ (𝐹𝐶))
71 f1of 6843 . . . . . . . 8 (𝐺:𝐶1-1-onto𝐷𝐺:𝐶𝐷)
72713ad2ant2 1132 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) → 𝐺:𝐶𝐷)
7372adantr 480 . . . . . 6 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → 𝐺:𝐶𝐷)
74 eqid 2733 . . . . . 6 (𝐺 ↾ (ran 𝐹𝐶)) = (𝐺 ↾ (ran 𝐹𝐶))
7567, 68, 69, 70, 73, 74fcoresf1ob 46973 . . . . 5 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → ((𝐺𝐹):(𝐹𝐶)–1-1-onto𝐷 ↔ ((𝐹 ↾ (𝐹𝐶)):(𝐹𝐶)–1-1→(ran 𝐹𝐶) ∧ (𝐺 ↾ (ran 𝐹𝐶)):(ran 𝐹𝐶)–1-1-onto𝐷)))
7664, 75bitrd 279 . . . 4 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → ((𝐺𝐹):(𝐹𝐶)–1-1-onto𝐷 ↔ ((𝐹 ↾ (𝐹𝐶)):(𝐹𝐶)–1-1→(ran 𝐹𝐶) ∧ (𝐺 ↾ (ran 𝐹𝐶)):(ran 𝐹𝐶)–1-1-onto𝐷)))
7724, 56, 76mpbir2and 712 . . 3 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → (𝐺𝐹):(𝐹𝐶)–1-1-onto𝐷)
78 simpl3 1191 . . 3 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → 𝐻:𝐸1-1-onto𝐼)
79 simprr 772 . . 3 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → 𝐷𝐸)
80 f1ocoima 7316 . . 3 (((𝐺𝐹):(𝐹𝐶)–1-1-onto𝐷𝐻:𝐸1-1-onto𝐼𝐷𝐸) → (𝐻 ∘ (𝐺𝐹)):(𝐹𝐶)–1-1-onto→(𝐻𝐷))
8177, 78, 79, 80syl3anc 1369 . 2 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → (𝐻 ∘ (𝐺𝐹)):(𝐹𝐶)–1-1-onto→(𝐻𝐷))
82 coass 6281 . . 3 ((𝐻𝐺) ∘ 𝐹) = (𝐻 ∘ (𝐺𝐹))
83 f1oeq1 6831 . . 3 (((𝐻𝐺) ∘ 𝐹) = (𝐻 ∘ (𝐺𝐹)) → (((𝐻𝐺) ∘ 𝐹):(𝐹𝐶)–1-1-onto→(𝐻𝐷) ↔ (𝐻 ∘ (𝐺𝐹)):(𝐹𝐶)–1-1-onto→(𝐻𝐷)))
8482, 83ax-mp 5 . 2 (((𝐻𝐺) ∘ 𝐹):(𝐹𝐶)–1-1-onto→(𝐻𝐷) ↔ (𝐻 ∘ (𝐺𝐹)):(𝐹𝐶)–1-1-onto→(𝐻𝐷))
8581, 84sylibr 234 1 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐴𝐷𝐸)) → ((𝐻𝐺) ∘ 𝐹):(𝐹𝐶)–1-1-onto→(𝐻𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1085   = wceq 1535  cin 3962  wss 3963  ccnv 5682  dom cdm 5683  ran crn 5684  cres 5685  cima 5686  ccom 5687  Rel wrel 5688   Fn wfn 6553  wf 6554  1-1wf1 6555  ontowfo 6556  1-1-ontowf1o 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6510  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566
This theorem is referenced by:  3f1oss2  46976  uspgrlim  47817
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