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Theorem ovmpt2df 5555
Description: Alternate deduction version of ovmpt2 5559, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
ovmpt2df.1 (φA 𝐶)
ovmpt2df.2 ((φ x = A) → B 𝐷)
ovmpt2df.3 ((φ (x = A y = B)) → 𝑅 𝑉)
ovmpt2df.4 ((φ (x = A y = B)) → ((A𝐹B) = 𝑅ψ))
ovmpt2df.5 x𝐹
ovmpt2df.6 xψ
ovmpt2df.7 y𝐹
ovmpt2df.8 yψ
Assertion
Ref Expression
ovmpt2df (φ → (𝐹 = (x 𝐶, y 𝐷𝑅) → ψ))
Distinct variable groups:   x,y,A   y,B   φ,x,y
Allowed substitution hints:   ψ(x,y)   B(x)   𝐶(x,y)   𝐷(x,y)   𝑅(x,y)   𝐹(x,y)   𝑉(x,y)

Proof of Theorem ovmpt2df
StepHypRef Expression
1 nfv 1402 . 2 xφ
2 ovmpt2df.5 . . . 4 x𝐹
3 nfmpt21 5494 . . . 4 x(x 𝐶, y 𝐷𝑅)
42, 3nfeq 2167 . . 3 x 𝐹 = (x 𝐶, y 𝐷𝑅)
5 ovmpt2df.6 . . 3 xψ
64, 5nfim 1446 . 2 x(𝐹 = (x 𝐶, y 𝐷𝑅) → ψ)
7 ovmpt2df.1 . . . 4 (φA 𝐶)
8 elex 2543 . . . 4 (A 𝐶A V)
97, 8syl 14 . . 3 (φA V)
10 isset 2539 . . 3 (A V ↔ x x = A)
119, 10sylib 127 . 2 (φx x = A)
12 ovmpt2df.2 . . . . 5 ((φ x = A) → B 𝐷)
13 elex 2543 . . . . 5 (B 𝐷B V)
1412, 13syl 14 . . . 4 ((φ x = A) → B V)
15 isset 2539 . . . 4 (B V ↔ y y = B)
1614, 15sylib 127 . . 3 ((φ x = A) → y y = B)
17 nfv 1402 . . . 4 y(φ x = A)
18 ovmpt2df.7 . . . . . 6 y𝐹
19 nfmpt22 5495 . . . . . 6 y(x 𝐶, y 𝐷𝑅)
2018, 19nfeq 2167 . . . . 5 y 𝐹 = (x 𝐶, y 𝐷𝑅)
21 ovmpt2df.8 . . . . 5 yψ
2220, 21nfim 1446 . . . 4 y(𝐹 = (x 𝐶, y 𝐷𝑅) → ψ)
23 oveq 5442 . . . . . 6 (𝐹 = (x 𝐶, y 𝐷𝑅) → (A𝐹B) = (A(x 𝐶, y 𝐷𝑅)B))
24 simprl 471 . . . . . . . . . 10 ((φ (x = A y = B)) → x = A)
25 simprr 472 . . . . . . . . . 10 ((φ (x = A y = B)) → y = B)
2624, 25oveq12d 5454 . . . . . . . . 9 ((φ (x = A y = B)) → (x(x 𝐶, y 𝐷𝑅)y) = (A(x 𝐶, y 𝐷𝑅)B))
277adantr 261 . . . . . . . . . . 11 ((φ (x = A y = B)) → A 𝐶)
2824, 27eqeltrd 2096 . . . . . . . . . 10 ((φ (x = A y = B)) → x 𝐶)
2912adantrr 451 . . . . . . . . . . 11 ((φ (x = A y = B)) → B 𝐷)
3025, 29eqeltrd 2096 . . . . . . . . . 10 ((φ (x = A y = B)) → y 𝐷)
31 ovmpt2df.3 . . . . . . . . . 10 ((φ (x = A y = B)) → 𝑅 𝑉)
32 eqid 2022 . . . . . . . . . . 11 (x 𝐶, y 𝐷𝑅) = (x 𝐶, y 𝐷𝑅)
3332ovmpt4g 5546 . . . . . . . . . 10 ((x 𝐶 y 𝐷 𝑅 𝑉) → (x(x 𝐶, y 𝐷𝑅)y) = 𝑅)
3428, 30, 31, 33syl3anc 1121 . . . . . . . . 9 ((φ (x = A y = B)) → (x(x 𝐶, y 𝐷𝑅)y) = 𝑅)
3526, 34eqtr3d 2056 . . . . . . . 8 ((φ (x = A y = B)) → (A(x 𝐶, y 𝐷𝑅)B) = 𝑅)
3635eqeq2d 2033 . . . . . . 7 ((φ (x = A y = B)) → ((A𝐹B) = (A(x 𝐶, y 𝐷𝑅)B) ↔ (A𝐹B) = 𝑅))
37 ovmpt2df.4 . . . . . . 7 ((φ (x = A y = B)) → ((A𝐹B) = 𝑅ψ))
3836, 37sylbid 139 . . . . . 6 ((φ (x = A y = B)) → ((A𝐹B) = (A(x 𝐶, y 𝐷𝑅)B) → ψ))
3923, 38syl5 28 . . . . 5 ((φ (x = A y = B)) → (𝐹 = (x 𝐶, y 𝐷𝑅) → ψ))
4039expr 357 . . . 4 ((φ x = A) → (y = B → (𝐹 = (x 𝐶, y 𝐷𝑅) → ψ)))
4117, 22, 40exlimd 1470 . . 3 ((φ x = A) → (y y = B → (𝐹 = (x 𝐶, y 𝐷𝑅) → ψ)))
4216, 41mpd 13 . 2 ((φ x = A) → (𝐹 = (x 𝐶, y 𝐷𝑅) → ψ))
431, 6, 11, 42exlimdd 1734 1 (φ → (𝐹 = (x 𝐶, y 𝐷𝑅) → ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1228  wnf 1329  wex 1362   wcel 1374  wnfc 2147  Vcvv 2535  (class class class)co 5436  cmpt2 5438
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-in1 532  ax-in2 533  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-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-setind 4204
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  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-ne 2188  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-dif 2897  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-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-iota 4794  df-fun 4831  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441
This theorem is referenced by:  ovmpt2dv  5556  ovmpt2dv2  5557
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