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Theorem mpt2fvex 5829
 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 5515 . 2 (𝑅𝐹𝑆) = (𝐹‘⟨𝑅, 𝑆⟩)
2 elex 2566 . . . . . . . . 9 (𝐶𝑉𝐶 ∈ V)
32alimi 1344 . . . . . . . 8 (∀𝑦 𝐶𝑉 → ∀𝑦 𝐶 ∈ V)
4 vex 2560 . . . . . . . . 9 𝑧 ∈ V
5 2ndexg 5795 . . . . . . . . 9 (𝑧 ∈ V → (2nd𝑧) ∈ V)
6 nfcv 2178 . . . . . . . . . 10 𝑦(2nd𝑧)
7 nfcsb1v 2882 . . . . . . . . . . 11 𝑦(2nd𝑧) / 𝑦𝐶
87nfel1 2188 . . . . . . . . . 10 𝑦(2nd𝑧) / 𝑦𝐶 ∈ V
9 csbeq1a 2860 . . . . . . . . . . 11 (𝑦 = (2nd𝑧) → 𝐶 = (2nd𝑧) / 𝑦𝐶)
109eleq1d 2106 . . . . . . . . . 10 (𝑦 = (2nd𝑧) → (𝐶 ∈ V ↔ (2nd𝑧) / 𝑦𝐶 ∈ V))
116, 8, 10spcgf 2635 . . . . . . . . 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 1344 . . . . . 6 (∀𝑥𝑦 𝐶𝑉 → ∀𝑥(2nd𝑧) / 𝑦𝐶 ∈ V)
15 1stexg 5794 . . . . . . 7 (𝑧 ∈ V → (1st𝑧) ∈ V)
16 nfcv 2178 . . . . . . . 8 𝑥(1st𝑧)
17 nfcsb1v 2882 . . . . . . . . 9 𝑥(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶
1817nfel1 2188 . . . . . . . 8 𝑥(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V
19 csbeq1a 2860 . . . . . . . . 9 (𝑥 = (1st𝑧) → (2nd𝑧) / 𝑦𝐶 = (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
2019eleq1d 2106 . . . . . . . 8 (𝑥 = (1st𝑧) → ((2nd𝑧) / 𝑦𝐶 ∈ V ↔ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V))
2116, 18, 20spcgf 2635 . . . . . . 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 1754 . . . 4 (∀𝑥𝑦 𝐶𝑉 → ∀𝑧(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V)
25243ad2ant1 925 . . 3 ((∀𝑥𝑦 𝐶𝑉𝑅𝑊𝑆𝑋) → ∀𝑧(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V)
26 opexg 3964 . . . 4 ((𝑅𝑊𝑆𝑋) → ⟨𝑅, 𝑆⟩ ∈ V)
27263adant1 922 . . 3 ((∀𝑥𝑦 𝐶𝑉𝑅𝑊𝑆𝑋) → ⟨𝑅, 𝑆⟩ ∈ V)
28 fmpt2.1 . . . . 5 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
29 mpt2mptsx 5823 . . . . 5 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
3028, 29eqtri 2060 . . . 4 𝐹 = (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
3130mptfvex 5256 . . 3 ((∀𝑧(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V ∧ ⟨𝑅, 𝑆⟩ ∈ V) → (𝐹‘⟨𝑅, 𝑆⟩) ∈ V)
3225, 27, 31syl2anc 391 . 2 ((∀𝑥𝑦 𝐶𝑉𝑅𝑊𝑆𝑋) → (𝐹‘⟨𝑅, 𝑆⟩) ∈ V)
331, 32syl5eqel 2124 1 ((∀𝑥𝑦 𝐶𝑉𝑅𝑊𝑆𝑋) → (𝑅𝐹𝑆) ∈ V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ w3a 885  ∀wal 1241   = wceq 1243   ∈ wcel 1393  Vcvv 2557  ⦋csb 2852  {csn 3375  ⟨cop 3378  ∪ ciun 3657   ↦ cmpt 3818   × cxp 4343  ‘cfv 4902  (class class class)co 5512   ↦ cmpt2 5514  1st c1st 5765  2nd c2nd 5766 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fo 4908  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768 This theorem is referenced by:  mpt2fvexi  5832  oaexg  6028  omexg  6031  oeiexg  6033
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