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Theorem cnvf1olem 8055

Description: Lemma for cnvf1o 8056. (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 773	. . . . 5 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐶 = ∪ ^◡{𝐵})
2		1st2nd 7986	. . . . . . . . 9 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
3	2	adantrr 718	. . . . . . . 8 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
4	3	sneqd 4593	. . . . . . 7 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → {𝐵} = {⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩})
5	4	cnveqd 5825	. . . . . 6 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ^◡{𝐵} = ^◡{⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩})
6	5	unieqd 4877	. . . . 5 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ∪ ^◡{𝐵} = ∪ ^◡{⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩})
7	1, 6	eqtrd 2772	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐶 = ∪ ^◡{⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩})
8		opswap 6188	. . . 4 ⊢ ∪ ^◡{⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩} = ⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩
9	7, 8	eqtrdi 2788	. . 3 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐶 = ⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩)
10		simprl 771	. . . . 5 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐵 ∈ 𝐴)
11	3, 10	eqeltrrd 2838	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ 𝐴)
12		fvex 6848	. . . . 5 ⊢ (2^nd ‘𝐵) ∈ V
13		fvex 6848	. . . . 5 ⊢ (1^st ‘𝐵) ∈ V
14	12, 13	opelcnv 5831	. . . 4 ⊢ (⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩ ∈ ^◡𝐴 ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ 𝐴)
15	11, 14	sylibr 234	. . 3 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩ ∈ ^◡𝐴)
16	9, 15	eqeltrd 2837	. 2 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐶 ∈ ^◡𝐴)
17		opswap 6188	. . . 4 ⊢ ∪ ^◡{⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩} = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩
18	17	eqcomi 2746	. . 3 ⊢ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ = ∪ ^◡{⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩}
19	9	sneqd 4593	. . . . 5 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → {𝐶} = {⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩})
20	19	cnveqd 5825	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ^◡{𝐶} = ^◡{⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩})
21	20	unieqd 4877	. . 3 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ∪ ^◡{𝐶} = ∪ ^◡{⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩})
22	18, 3, 21	3eqtr4a 2798	. 2 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐵 = ∪ ^◡{𝐶})
23	16, 22	jca 511	1 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → (𝐶 ∈ ^◡𝐴 ∧ 𝐵 = ∪ ^◡{𝐶}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 {csn 4581 ⟨cop 4587 ∪ cuni 4864 ^◡ccnv 5624 Rel wrel 5630 ‘cfv 6493 1^st c1st 7934 2^nd c2nd 7935

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5242 ax-nul 5252 ax-pr 5378 ax-un 7683

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-rab 3401 df-v 3443 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4287 df-if 4481 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-iota 6449 df-fun 6495 df-fv 6501 df-1st 7936 df-2nd 7937

This theorem is referenced by: cnvf1o 8056 fcnvgreu 32754 gsumhashmul 33153

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