Theorem fparlem2 7808

Description: Lemma for fpar 7811. (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 6689	. . . . . 6 ⊢ (𝑥 ∈ (V × V) → ((2nd ↾ (V × V))‘𝑥) = (2nd ‘𝑥))
2	1	eqeq1d 2823	. . . . 5 ⊢ (𝑥 ∈ (V × V) → (((2nd ↾ (V × V))‘𝑥) = 𝑦 ↔ (2nd ‘𝑥) = 𝑦))
3		vex 3497	. . . . . . 7 ⊢ 𝑦 ∈ V
4	3	elsn2 4604	. . . . . 6 ⊢ ((2nd ‘𝑥) ∈ {𝑦} ↔ (2nd ‘𝑥) = 𝑦)
5		fvex 6683	. . . . . . 7 ⊢ (1st ‘𝑥) ∈ V
6	5	biantrur 533	. . . . . 6 ⊢ ((2nd ‘𝑥) ∈ {𝑦} ↔ ((1st ‘𝑥) ∈ V ∧ (2nd ‘𝑥) ∈ {𝑦}))
7	4, 6	bitr3i 279	. . . . 5 ⊢ ((2nd ‘𝑥) = 𝑦 ↔ ((1st ‘𝑥) ∈ V ∧ (2nd ‘𝑥) ∈ {𝑦}))
8	2, 7	syl6bb 289	. . . 4 ⊢ (𝑥 ∈ (V × V) → (((2nd ↾ (V × V))‘𝑥) = 𝑦 ↔ ((1st ‘𝑥) ∈ V ∧ (2nd ‘𝑥) ∈ {𝑦})))
9	8	pm5.32i 577	. . 3 ⊢ ((𝑥 ∈ (V × V) ∧ ((2nd ↾ (V × V))‘𝑥) = 𝑦) ↔ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ V ∧ (2nd ‘𝑥) ∈ {𝑦})))
10		f2ndres 7714	. . . 4 ⊢ (2nd ↾ (V × V)):(V × V)⟶V
11		ffn 6514	. . . 4 ⊢ ((2nd ↾ (V × V)):(V × V)⟶V → (2nd ↾ (V × V)) Fn (V × V))
12		fniniseg 6830	. . . 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 7724	. . 3 ⊢ (𝑥 ∈ (V × {𝑦}) ↔ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ V ∧ (2nd ‘𝑥) ∈ {𝑦})))
15	9, 13, 14	3bitr4i 305	. 2 ⊢ (𝑥 ∈ (◡(2nd ↾ (V × V)) “ {𝑦}) ↔ 𝑥 ∈ (V × {𝑦}))
16	15	eqriv 2818	1 ⊢ (◡(2nd ↾ (V × V)) “ {𝑦}) = (V × {𝑦})

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3494  {csn 4567   × cxp 5553  ^◡ccnv 5554   ↾ cres 5557   “ cima 5558   Fn wfn 6350  ⟶wf 6351  ‘cfv 6355  1^st c1st 7687  2^nd c2nd 7688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fv 6363  df-1st 7689  df-2nd 7690

This theorem is referenced by:  fparlem4  7810