Theorem fparlem2 8098

Description: Lemma for fpar 8101. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
fparlem2	⊢ (◡(2nd ↾ (V × V)) “ {𝑦}) = (V × {𝑦})

Proof of Theorem fparlem2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvres 6910	. . . . . 6 ⊢ (𝑥 ∈ (V × V) → ((2nd ↾ (V × V))‘𝑥) = (2nd ‘𝑥))
2	1	eqeq1d 2734	. . . . 5 ⊢ (𝑥 ∈ (V × V) → (((2nd ↾ (V × V))‘𝑥) = 𝑦 ↔ (2nd ‘𝑥) = 𝑦))
3		vex 3478	. . . . . . 7 ⊢ 𝑦 ∈ V
4	3	elsn2 4667	. . . . . 6 ⊢ ((2nd ‘𝑥) ∈ {𝑦} ↔ (2nd ‘𝑥) = 𝑦)
5		fvex 6904	. . . . . . 7 ⊢ (1st ‘𝑥) ∈ V
6	5	biantrur 531	. . . . . 6 ⊢ ((2nd ‘𝑥) ∈ {𝑦} ↔ ((1st ‘𝑥) ∈ V ∧ (2nd ‘𝑥) ∈ {𝑦}))
7	4, 6	bitr3i 276	. . . . 5 ⊢ ((2nd ‘𝑥) = 𝑦 ↔ ((1st ‘𝑥) ∈ V ∧ (2nd ‘𝑥) ∈ {𝑦}))
8	2, 7	bitrdi 286	. . . 4 ⊢ (𝑥 ∈ (V × V) → (((2nd ↾ (V × V))‘𝑥) = 𝑦 ↔ ((1st ‘𝑥) ∈ V ∧ (2nd ‘𝑥) ∈ {𝑦})))
9	8	pm5.32i 575	. . 3 ⊢ ((𝑥 ∈ (V × V) ∧ ((2nd ↾ (V × V))‘𝑥) = 𝑦) ↔ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ V ∧ (2nd ‘𝑥) ∈ {𝑦})))
10		f2ndres 7999	. . . 4 ⊢ (2nd ↾ (V × V)):(V × V)⟶V
11		ffn 6717	. . . 4 ⊢ ((2nd ↾ (V × V)):(V × V)⟶V → (2nd ↾ (V × V)) Fn (V × V))
12		fniniseg 7061	. . . 4 ⊢ ((2nd ↾ (V × V)) Fn (V × V) → (𝑥 ∈ (◡(2nd ↾ (V × V)) “ {𝑦}) ↔ (𝑥 ∈ (V × V) ∧ ((2nd ↾ (V × V))‘𝑥) = 𝑦)))
13	10, 11, 12	mp2b 10	. . 3 ⊢ (𝑥 ∈ (◡(2nd ↾ (V × V)) “ {𝑦}) ↔ (𝑥 ∈ (V × V) ∧ ((2nd ↾ (V × V))‘𝑥) = 𝑦))
14		elxp7 8009	. . 3 ⊢ (𝑥 ∈ (V × {𝑦}) ↔ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ V ∧ (2nd ‘𝑥) ∈ {𝑦})))
15	9, 13, 14	3bitr4i 302	. 2 ⊢ (𝑥 ∈ (◡(2nd ↾ (V × V)) “ {𝑦}) ↔ 𝑥 ∈ (V × {𝑦}))
16	15	eqriv 2729	1 ⊢ (◡(2nd ↾ (V × V)) “ {𝑦}) = (V × {𝑦})

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474  {csn 4628   × cxp 5674  ^◡ccnv 5675   ↾ cres 5678   “ cima 5679   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  1^st c1st 7972  2^nd c2nd 7973

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-1st 7974  df-2nd 7975

This theorem is referenced by:  fparlem4  8100