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Theorem cnvf1olem 8110

Description: Lemma for cnvf1o 8111. (Contributed by Mario Carneiro, 27-Apr-2014.)

**Assertion**
Ref	Expression
cnvf1olem	⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → (𝐶 ∈ ^◡𝐴 ∧ 𝐵 = ∪ ^◡{𝐶}))

**Proof of Theorem cnvf1olem**
Step	Hyp	Ref	Expression
1		simprr 785	. . . . 5 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐶 = ∪ ^◡{𝐵})
2		1st2nd 8039	. . . . . . . . 9 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
3	2	adantrr 730	. . . . . . . 8 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
4	3	sneqd 4599	. . . . . . 7 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → {𝐵} = {⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩})
5	4	cnveqd 5859	. . . . . 6 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ^◡{𝐵} = ^◡{⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩})
6	5	unieqd 4883	. . . . 5 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ∪ ^◡{𝐵} = ∪ ^◡{⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩})
7	1, 6	eqtrd 2797	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐶 = ∪ ^◡{⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩})
8		opswap 6229	. . . 4 ⊢ ∪ ^◡{⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩} = ⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩
9	7, 8	eqtrdi 2813	. . 3 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐶 = ⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩)
10		simprl 783	. . . . 5 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐵 ∈ 𝐴)
11	3, 10	eqeltrrd 2863	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ 𝐴)
12		fvex 6895	. . . . 5 ⊢ (2^nd ‘𝐵) ∈ V
13		fvex 6895	. . . . 5 ⊢ (1^st ‘𝐵) ∈ V
14	12, 13	opelcnv 5865	. . . 4 ⊢ (⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩ ∈ ^◡𝐴 ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ 𝐴)
15	11, 14	sylibr 237	. . 3 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩ ∈ ^◡𝐴)
16	9, 15	eqeltrd 2862	. 2 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐶 ∈ ^◡𝐴)
17		opswap 6229	. . . 4 ⊢ ∪ ^◡{⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩} = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩
18	17	eqcomi 2771	. . 3 ⊢ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ = ∪ ^◡{⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩}
19	9	sneqd 4599	. . . . 5 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → {𝐶} = {⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩})
20	19	cnveqd 5859	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ^◡{𝐶} = ^◡{⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩})
21	20	unieqd 4883	. . 3 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ∪ ^◡{𝐶} = ∪ ^◡{⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩})
22	18, 3, 21	3eqtr4a 2823	. 2 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐵 = ∪ ^◡{𝐶})
23	16, 22	jca 521	1 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → (𝐶 ∈ ^◡𝐴 ∧ 𝐵 = ∪ ^◡{𝐶}))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 {csn 4587 ⟨cop 4593 ∪ cuni 4870 ^◡ccnv 5658 Rel wrel 5664 ‘cfv 6537 1^st c1st 7987 2^nd c2nd 7988

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pr 5402 ax-un 7739

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-rab 3415 df-v 3455 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-br 5108 df-opab 5172 df-mpt 5191 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-iota 6493 df-fun 6539 df-fv 6545 df-1st 7989 df-2nd 7990

This theorem is used by: cnvf1o 8111 fcnvgreu 33132 gsumhashmul 33494

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