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Theorem mpt2fvex 5906
Description: Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
Hypothesis
Ref Expression
fmpt2.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
mpt2fvex ((∀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 mpt2fvex
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-ov 5592 . 2 (𝑅𝐹𝑆) = (𝐹‘⟨𝑅, 𝑆⟩)
2 elex 2621 . . . . . . . . 9 (𝐶𝑉𝐶 ∈ V)
32alimi 1385 . . . . . . . 8 (∀𝑦 𝐶𝑉 → ∀𝑦 𝐶 ∈ V)
4 vex 2615 . . . . . . . . 9 𝑧 ∈ V
5 2ndexg 5872 . . . . . . . . 9 (𝑧 ∈ V → (2nd𝑧) ∈ V)
6 nfcv 2223 . . . . . . . . . 10 𝑦(2nd𝑧)
7 nfcsb1v 2949 . . . . . . . . . . 11 𝑦(2nd𝑧) / 𝑦𝐶
87nfel1 2233 . . . . . . . . . 10 𝑦(2nd𝑧) / 𝑦𝐶 ∈ V
9 csbeq1a 2927 . . . . . . . . . . 11 (𝑦 = (2nd𝑧) → 𝐶 = (2nd𝑧) / 𝑦𝐶)
109eleq1d 2151 . . . . . . . . . 10 (𝑦 = (2nd𝑧) → (𝐶 ∈ V ↔ (2nd𝑧) / 𝑦𝐶 ∈ V))
116, 8, 10spcgf 2691 . . . . . . . . 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 1385 . . . . . 6 (∀𝑥𝑦 𝐶𝑉 → ∀𝑥(2nd𝑧) / 𝑦𝐶 ∈ V)
15 1stexg 5871 . . . . . . 7 (𝑧 ∈ V → (1st𝑧) ∈ V)
16 nfcv 2223 . . . . . . . 8 𝑥(1st𝑧)
17 nfcsb1v 2949 . . . . . . . . 9 𝑥(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶
1817nfel1 2233 . . . . . . . 8 𝑥(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V
19 csbeq1a 2927 . . . . . . . . 9 (𝑥 = (1st𝑧) → (2nd𝑧) / 𝑦𝐶 = (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
2019eleq1d 2151 . . . . . . . 8 (𝑥 = (1st𝑧) → ((2nd𝑧) / 𝑦𝐶 ∈ V ↔ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V))
2116, 18, 20spcgf 2691 . . . . . . 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 1797 . . . 4 (∀𝑥𝑦 𝐶𝑉 → ∀𝑧(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V)
25243ad2ant1 960 . . 3 ((∀𝑥𝑦 𝐶𝑉𝑅𝑊𝑆𝑋) → ∀𝑧(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V)
26 opexg 4018 . . . 4 ((𝑅𝑊𝑆𝑋) → ⟨𝑅, 𝑆⟩ ∈ V)
27263adant1 957 . . 3 ((∀𝑥𝑦 𝐶𝑉𝑅𝑊𝑆𝑋) → ⟨𝑅, 𝑆⟩ ∈ V)
28 fmpt2.1 . . . . 5 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
29 mpt2mptsx 5900 . . . . 5 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
3028, 29eqtri 2103 . . . 4 𝐹 = (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
3130mptfvex 5331 . . 3 ((∀𝑧(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V ∧ ⟨𝑅, 𝑆⟩ ∈ V) → (𝐹‘⟨𝑅, 𝑆⟩) ∈ V)
3225, 27, 31syl2anc 403 . 2 ((∀𝑥𝑦 𝐶𝑉𝑅𝑊𝑆𝑋) → (𝐹‘⟨𝑅, 𝑆⟩) ∈ V)
331, 32syl5eqel 2169 1 ((∀𝑥𝑦 𝐶𝑉𝑅𝑊𝑆𝑋) → (𝑅𝐹𝑆) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 920  wal 1283   = wceq 1285  wcel 1434  Vcvv 2612  csb 2919  {csn 3422  cop 3425   ciun 3704  cmpt 3865   × cxp 4397  cfv 4967  (class class class)co 5589  cmpt2 5591  1st c1st 5842  2nd c2nd 5843
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999  ax-un 4223
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-csb 2920  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4083  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-fo 4973  df-fv 4975  df-ov 5592  df-oprab 5593  df-mpt2 5594  df-1st 5844  df-2nd 5845
This theorem is referenced by:  mpt2fvexi  5909  oaexg  6139  omexg  6142  oeiexg  6144
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