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Theorem mpofvex 6203
Description: Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
Hypothesis
Ref Expression
fmpo.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
mpofvex ((∀𝑥𝑦 𝐶𝑉𝑅𝑊𝑆𝑋) → (𝑅𝐹𝑆) ∈ V)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem mpofvex
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-ov 5877 . 2 (𝑅𝐹𝑆) = (𝐹‘⟨𝑅, 𝑆⟩)
2 elex 2748 . . . . . . . . 9 (𝐶𝑉𝐶 ∈ V)
32alimi 1455 . . . . . . . 8 (∀𝑦 𝐶𝑉 → ∀𝑦 𝐶 ∈ V)
4 vex 2740 . . . . . . . . 9 𝑧 ∈ V
5 2ndexg 6168 . . . . . . . . 9 (𝑧 ∈ V → (2nd𝑧) ∈ V)
6 nfcv 2319 . . . . . . . . . 10 𝑦(2nd𝑧)
7 nfcsb1v 3090 . . . . . . . . . . 11 𝑦(2nd𝑧) / 𝑦𝐶
87nfel1 2330 . . . . . . . . . 10 𝑦(2nd𝑧) / 𝑦𝐶 ∈ V
9 csbeq1a 3066 . . . . . . . . . . 11 (𝑦 = (2nd𝑧) → 𝐶 = (2nd𝑧) / 𝑦𝐶)
109eleq1d 2246 . . . . . . . . . 10 (𝑦 = (2nd𝑧) → (𝐶 ∈ V ↔ (2nd𝑧) / 𝑦𝐶 ∈ V))
116, 8, 10spcgf 2819 . . . . . . . . 9 ((2nd𝑧) ∈ V → (∀𝑦 𝐶 ∈ V → (2nd𝑧) / 𝑦𝐶 ∈ V))
124, 5, 11mp2b 8 . . . . . . . 8 (∀𝑦 𝐶 ∈ V → (2nd𝑧) / 𝑦𝐶 ∈ V)
133, 12syl 14 . . . . . . 7 (∀𝑦 𝐶𝑉(2nd𝑧) / 𝑦𝐶 ∈ V)
1413alimi 1455 . . . . . 6 (∀𝑥𝑦 𝐶𝑉 → ∀𝑥(2nd𝑧) / 𝑦𝐶 ∈ V)
15 1stexg 6167 . . . . . . 7 (𝑧 ∈ V → (1st𝑧) ∈ V)
16 nfcv 2319 . . . . . . . 8 𝑥(1st𝑧)
17 nfcsb1v 3090 . . . . . . . . 9 𝑥(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶
1817nfel1 2330 . . . . . . . 8 𝑥(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V
19 csbeq1a 3066 . . . . . . . . 9 (𝑥 = (1st𝑧) → (2nd𝑧) / 𝑦𝐶 = (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
2019eleq1d 2246 . . . . . . . 8 (𝑥 = (1st𝑧) → ((2nd𝑧) / 𝑦𝐶 ∈ V ↔ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V))
2116, 18, 20spcgf 2819 . . . . . . 7 ((1st𝑧) ∈ V → (∀𝑥(2nd𝑧) / 𝑦𝐶 ∈ V → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V))
224, 15, 21mp2b 8 . . . . . 6 (∀𝑥(2nd𝑧) / 𝑦𝐶 ∈ V → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V)
2314, 22syl 14 . . . . 5 (∀𝑥𝑦 𝐶𝑉(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V)
2423alrimiv 1874 . . . 4 (∀𝑥𝑦 𝐶𝑉 → ∀𝑧(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V)
25243ad2ant1 1018 . . 3 ((∀𝑥𝑦 𝐶𝑉𝑅𝑊𝑆𝑋) → ∀𝑧(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V)
26 opexg 4228 . . . 4 ((𝑅𝑊𝑆𝑋) → ⟨𝑅, 𝑆⟩ ∈ V)
27263adant1 1015 . . 3 ((∀𝑥𝑦 𝐶𝑉𝑅𝑊𝑆𝑋) → ⟨𝑅, 𝑆⟩ ∈ V)
28 fmpo.1 . . . . 5 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
29 mpomptsx 6197 . . . . 5 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
3028, 29eqtri 2198 . . . 4 𝐹 = (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
3130mptfvex 5601 . . 3 ((∀𝑧(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V ∧ ⟨𝑅, 𝑆⟩ ∈ V) → (𝐹‘⟨𝑅, 𝑆⟩) ∈ V)
3225, 27, 31syl2anc 411 . 2 ((∀𝑥𝑦 𝐶𝑉𝑅𝑊𝑆𝑋) → (𝐹‘⟨𝑅, 𝑆⟩) ∈ V)
331, 32eqeltrid 2264 1 ((∀𝑥𝑦 𝐶𝑉𝑅𝑊𝑆𝑋) → (𝑅𝐹𝑆) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 978  wal 1351   = wceq 1353  wcel 2148  Vcvv 2737  csb 3057  {csn 3592  cop 3595   ciun 3886  cmpt 4064   × cxp 4624  cfv 5216  (class class class)co 5874  cmpo 5876  1st c1st 6138  2nd c2nd 6139
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fo 5222  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141
This theorem is referenced by:  mpofvexi  6206  oaexg  6448  omexg  6451  oeiexg  6453
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