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Theorem mpt2fvex 5752
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 = (x A, y B𝐶)
Assertion
Ref Expression
mpt2fvex ((xy 𝐶 𝑉 𝑅 𝑊 𝑆 𝑋) → (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 5439 . 2 (𝑅𝐹𝑆) = (𝐹‘⟨𝑅, 𝑆⟩)
2 elex 2543 . . . . . . . . 9 (𝐶 𝑉𝐶 V)
32alimi 1324 . . . . . . . 8 (y 𝐶 𝑉y 𝐶 V)
4 vex 2538 . . . . . . . . 9 z V
5 2ndexg 5718 . . . . . . . . 9 (z V → (2ndz) V)
6 nfcv 2160 . . . . . . . . . 10 y(2ndz)
7 nfcsb1v 2859 . . . . . . . . . . 11 y(2ndz) / y𝐶
87nfel1 2170 . . . . . . . . . 10 y(2ndz) / y𝐶 V
9 csbeq1a 2837 . . . . . . . . . . 11 (y = (2ndz) → 𝐶 = (2ndz) / y𝐶)
109eleq1d 2088 . . . . . . . . . 10 (y = (2ndz) → (𝐶 V ↔ (2ndz) / y𝐶 V))
116, 8, 10spcgf 2612 . . . . . . . . 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 𝐶 𝑉(2ndz) / y𝐶 V)
1413alimi 1324 . . . . . 6 (xy 𝐶 𝑉x(2ndz) / y𝐶 V)
15 1stexg 5717 . . . . . . 7 (z V → (1stz) V)
16 nfcv 2160 . . . . . . . 8 x(1stz)
17 nfcsb1v 2859 . . . . . . . . 9 x(1stz) / x(2ndz) / y𝐶
1817nfel1 2170 . . . . . . . 8 x(1stz) / x(2ndz) / y𝐶 V
19 csbeq1a 2837 . . . . . . . . 9 (x = (1stz) → (2ndz) / y𝐶 = (1stz) / x(2ndz) / y𝐶)
2019eleq1d 2088 . . . . . . . 8 (x = (1stz) → ((2ndz) / y𝐶 V ↔ (1stz) / x(2ndz) / y𝐶 V))
2116, 18, 20spcgf 2612 . . . . . . 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 𝐶 𝑉(1stz) / x(2ndz) / y𝐶 V)
2423alrimiv 1736 . . . 4 (xy 𝐶 𝑉z(1stz) / x(2ndz) / y𝐶 V)
25243ad2ant1 913 . . 3 ((xy 𝐶 𝑉 𝑅 𝑊 𝑆 𝑋) → z(1stz) / x(2ndz) / y𝐶 V)
26 opexg 3938 . . . 4 ((𝑅 𝑊 𝑆 𝑋) → ⟨𝑅, 𝑆 V)
27263adant1 910 . . 3 ((xy 𝐶 𝑉 𝑅 𝑊 𝑆 𝑋) → ⟨𝑅, 𝑆 V)
28 fmpt2.1 . . . . 5 𝐹 = (x A, y B𝐶)
29 mpt2mptsx 5746 . . . . 5 (x A, y B𝐶) = (z x A ({x} × B) ↦ (1stz) / x(2ndz) / y𝐶)
3028, 29eqtri 2042 . . . 4 𝐹 = (z x A ({x} × B) ↦ (1stz) / x(2ndz) / y𝐶)
3130mptfvex 5181 . . 3 ((z(1stz) / x(2ndz) / y𝐶 V 𝑅, 𝑆 V) → (𝐹‘⟨𝑅, 𝑆⟩) V)
3225, 27, 31syl2anc 393 . 2 ((xy 𝐶 𝑉 𝑅 𝑊 𝑆 𝑋) → (𝐹‘⟨𝑅, 𝑆⟩) V)
331, 32syl5eqel 2106 1 ((xy 𝐶 𝑉 𝑅 𝑊 𝑆 𝑋) → (𝑅𝐹𝑆) V)
Colors of variables: wff set class
Syntax hints:  wi 4   w3a 873  wal 1226   = wceq 1228   wcel 1374  Vcvv 2535  csb 2829  {csn 3350  cop 3353   ciun 3631  cmpt 3792   × cxp 4270  cfv 4829  (class class class)co 5436  cmpt2 5438  1st c1st 5688  2nd c2nd 5689
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-fo 4835  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441  df-1st 5690  df-2nd 5691
This theorem is referenced by:  mpt2fvexi  5755  oaexg  5943  omexg  5946  oeiexg  5948
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