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Theorem ovmpt2df 5574
Description: Alternate deduction version of ovmpt2 5578, 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 1418 . 2 xφ
2 ovmpt2df.5 . . . 4 x𝐹
3 nfmpt21 5513 . . . 4 x(x 𝐶, y 𝐷𝑅)
42, 3nfeq 2182 . . 3 x 𝐹 = (x 𝐶, y 𝐷𝑅)
5 ovmpt2df.6 . . 3 xψ
64, 5nfim 1461 . 2 x(𝐹 = (x 𝐶, y 𝐷𝑅) → ψ)
7 ovmpt2df.1 . . . 4 (φA 𝐶)
8 elex 2560 . . . 4 (A 𝐶A V)
97, 8syl 14 . . 3 (φA V)
10 isset 2555 . . 3 (A V ↔ x x = A)
119, 10sylib 127 . 2 (φx x = A)
12 ovmpt2df.2 . . . . 5 ((φ x = A) → B 𝐷)
13 elex 2560 . . . . 5 (B 𝐷B V)
1412, 13syl 14 . . . 4 ((φ x = A) → B V)
15 isset 2555 . . . 4 (B V ↔ y y = B)
1614, 15sylib 127 . . 3 ((φ x = A) → y y = B)
17 nfv 1418 . . . 4 y(φ x = A)
18 ovmpt2df.7 . . . . . 6 y𝐹
19 nfmpt22 5514 . . . . . 6 y(x 𝐶, y 𝐷𝑅)
2018, 19nfeq 2182 . . . . 5 y 𝐹 = (x 𝐶, y 𝐷𝑅)
21 ovmpt2df.8 . . . . 5 yψ
2220, 21nfim 1461 . . . 4 y(𝐹 = (x 𝐶, y 𝐷𝑅) → ψ)
23 oveq 5461 . . . . . 6 (𝐹 = (x 𝐶, y 𝐷𝑅) → (A𝐹B) = (A(x 𝐶, y 𝐷𝑅)B))
24 simprl 483 . . . . . . . . . 10 ((φ (x = A y = B)) → x = A)
25 simprr 484 . . . . . . . . . 10 ((φ (x = A y = B)) → y = B)
2624, 25oveq12d 5473 . . . . . . . . 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 2111 . . . . . . . . . 10 ((φ (x = A y = B)) → x 𝐶)
2912adantrr 448 . . . . . . . . . . 11 ((φ (x = A y = B)) → B 𝐷)
3025, 29eqeltrd 2111 . . . . . . . . . 10 ((φ (x = A y = B)) → y 𝐷)
31 ovmpt2df.3 . . . . . . . . . 10 ((φ (x = A y = B)) → 𝑅 𝑉)
32 eqid 2037 . . . . . . . . . . 11 (x 𝐶, y 𝐷𝑅) = (x 𝐶, y 𝐷𝑅)
3332ovmpt4g 5565 . . . . . . . . . 10 ((x 𝐶 y 𝐷 𝑅 𝑉) → (x(x 𝐶, y 𝐷𝑅)y) = 𝑅)
3428, 30, 31, 33syl3anc 1134 . . . . . . . . 9 ((φ (x = A y = B)) → (x(x 𝐶, y 𝐷𝑅)y) = 𝑅)
3526, 34eqtr3d 2071 . . . . . . . 8 ((φ (x = A y = B)) → (A(x 𝐶, y 𝐷𝑅)B) = 𝑅)
3635eqeq2d 2048 . . . . . . 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 1485 . . 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 1749 1 (φ → (𝐹 = (x 𝐶, y 𝐷𝑅) → ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wnf 1346  wex 1378   wcel 1390  wnfc 2162  Vcvv 2551  (class class class)co 5455  cmpt2 5457
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460
This theorem is referenced by:  ovmpt2dv  5575  ovmpt2dv2  5576
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