Theorem mpofvexi 6432

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 1503	. 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 6431	. 2 ⊢ ((∀𝑥∀𝑦 𝐶 ∈ V ∧ 𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅𝐹𝑆) ∈ V)
7	2, 3, 4, 6	mp3an 1378	1 ⊢ (𝑅𝐹𝑆) ∈ V

Syntax hints:  ∀wal 1400   = wceq 1402   ∈ wcel 2209  Vcvv 2821  (class class class)co 6075   ∈ cmpo 6077

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fo 5378  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365

This theorem is referenced by:  metuex  14864  depindlem1  16661