Theorem elxp8 37866

Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp7 8006. (Contributed by ML, 19-Oct-2020.)

Assertion

Ref	Expression
elxp8	⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)))

Proof of Theorem elxp8

Step	Hyp	Ref	Expression
1		xp1st 8003	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → (1st ‘𝐴) ∈ 𝐵)
2		ssv 3961	. . . . 5 ⊢ 𝐵 ⊆ V
3		ssid 3959	. . . . 5 ⊢ 𝐶 ⊆ 𝐶
4		xpss12 5663	. . . . 5 ⊢ ((𝐵 ⊆ V ∧ 𝐶 ⊆ 𝐶) → (𝐵 × 𝐶) ⊆ (V × 𝐶))
5	2, 3, 4	mp2an 702	. . . 4 ⊢ (𝐵 × 𝐶) ⊆ (V × 𝐶)
6	5	sseli 3933	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ (V × 𝐶))
7	1, 6	jca 519	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)))
8		xpss 5664	. . . . 5 ⊢ (V × 𝐶) ⊆ (V × V)
9	8	sseli 3933	. . . 4 ⊢ (𝐴 ∈ (V × 𝐶) → 𝐴 ∈ (V × V))
10	9	adantl 485	. . 3 ⊢ (((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (V × V))
11		xp2nd 8004	. . . 4 ⊢ (𝐴 ∈ (V × 𝐶) → (2nd ‘𝐴) ∈ 𝐶)
12	11	anim2i 626	. . 3 ⊢ (((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))
13		elxp7 8006	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))
14	10, 12, 13	sylanbrc 592	. 2 ⊢ (((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (𝐵 × 𝐶))
15	7, 14	impbii 211	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)))

Syntax hints:   ↔ wb 208   ∧ wa 399   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3905   × cxp 5646  ‘cfv 6522  1^st c1st 7969  2^nd c2nd 7970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-iota 6478  df-fun 6524  df-fv 6530  df-1st 7971  df-2nd 7972

This theorem is referenced by:  finxpsuclem  37892