Theorem mpofvexi 6402

Description: Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)

Hypotheses

Ref	Expression
mpofvex.1	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
mpofvexi.c	⊢ 𝐶 ∈ V
mpofvexi.3	⊢ 𝑅 ∈ V
mpofvexi.4	⊢ 𝑆 ∈ V

Assertion

Ref	Expression
mpofvexi	⊢ (𝑅𝐹𝑆) ∈ V

Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem mpofvexi

Step	Hyp	Ref	Expression
1		mpofvexi.c	. . 3 ⊢ 𝐶 ∈ V
2	1	gen2 1499	. 2 ⊢ ∀𝑥∀𝑦 𝐶 ∈ V
3		mpofvexi.3	. 2 ⊢ 𝑅 ∈ V
4		mpofvexi.4	. 2 ⊢ 𝑆 ∈ V
5		mpofvex.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
6	5	mpofvex 6401	. 2 ⊢ ((∀𝑥∀𝑦 𝐶 ∈ V ∧ 𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅𝐹𝑆) ∈ V)
7	2, 3, 4, 6	mp3an 1374	1 ⊢ (𝑅𝐹𝑆) ∈ V

Syntax hints:  ∀wal 1396   = wceq 1398   ∈ wcel 2203  Vcvv 2813  (class class class)co 6050   ∈ cmpo 6052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fo 5358  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335

This theorem is referenced by:  metuex  14703  depindlem1  16501