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Theorem ovmpodf 5996
Description: Alternate deduction version of ovmpo 6000, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
ovmpodf.1 (𝜑𝐴𝐶)
ovmpodf.2 ((𝜑𝑥 = 𝐴) → 𝐵𝐷)
ovmpodf.3 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅𝑉)
ovmpodf.4 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = 𝑅𝜓))
ovmpodf.5 𝑥𝐹
ovmpodf.6 𝑥𝜓
ovmpodf.7 𝑦𝐹
ovmpodf.8 𝑦𝜓
Assertion
Ref Expression
ovmpodf (𝜑 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 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 ovmpodf
StepHypRef Expression
1 nfv 1526 . 2 𝑥𝜑
2 ovmpodf.5 . . . 4 𝑥𝐹
3 nfmpo1 5932 . . . 4 𝑥(𝑥𝐶, 𝑦𝐷𝑅)
42, 3nfeq 2325 . . 3 𝑥 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
5 ovmpodf.6 . . 3 𝑥𝜓
64, 5nfim 1570 . 2 𝑥(𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓)
7 ovmpodf.1 . . . 4 (𝜑𝐴𝐶)
8 elex 2746 . . . 4 (𝐴𝐶𝐴 ∈ V)
97, 8syl 14 . . 3 (𝜑𝐴 ∈ V)
10 isset 2741 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
119, 10sylib 122 . 2 (𝜑 → ∃𝑥 𝑥 = 𝐴)
12 ovmpodf.2 . . . . 5 ((𝜑𝑥 = 𝐴) → 𝐵𝐷)
13 elex 2746 . . . . 5 (𝐵𝐷𝐵 ∈ V)
1412, 13syl 14 . . . 4 ((𝜑𝑥 = 𝐴) → 𝐵 ∈ V)
15 isset 2741 . . . 4 (𝐵 ∈ V ↔ ∃𝑦 𝑦 = 𝐵)
1614, 15sylib 122 . . 3 ((𝜑𝑥 = 𝐴) → ∃𝑦 𝑦 = 𝐵)
17 nfv 1526 . . . 4 𝑦(𝜑𝑥 = 𝐴)
18 ovmpodf.7 . . . . . 6 𝑦𝐹
19 nfmpo2 5933 . . . . . 6 𝑦(𝑥𝐶, 𝑦𝐷𝑅)
2018, 19nfeq 2325 . . . . 5 𝑦 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
21 ovmpodf.8 . . . . 5 𝑦𝜓
2220, 21nfim 1570 . . . 4 𝑦(𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓)
23 oveq 5871 . . . . . 6 (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → (𝐴𝐹𝐵) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵))
24 simprl 529 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑥 = 𝐴)
25 simprr 531 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑦 = 𝐵)
2624, 25oveq12d 5883 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵))
277adantr 276 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝐴𝐶)
2824, 27eqeltrd 2252 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑥𝐶)
2912adantrr 479 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝐵𝐷)
3025, 29eqeltrd 2252 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑦𝐷)
31 ovmpodf.3 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅𝑉)
32 eqid 2175 . . . . . . . . . . 11 (𝑥𝐶, 𝑦𝐷𝑅) = (𝑥𝐶, 𝑦𝐷𝑅)
3332ovmpt4g 5987 . . . . . . . . . 10 ((𝑥𝐶𝑦𝐷𝑅𝑉) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)
3428, 30, 31, 33syl3anc 1238 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)
3526, 34eqtr3d 2210 . . . . . . . 8 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑅)
3635eqeq2d 2187 . . . . . . 7 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) ↔ (𝐴𝐹𝐵) = 𝑅))
37 ovmpodf.4 . . . . . . 7 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = 𝑅𝜓))
3836, 37sylbid 150 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) → 𝜓))
3923, 38syl5 32 . . . . 5 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓))
4039expr 375 . . . 4 ((𝜑𝑥 = 𝐴) → (𝑦 = 𝐵 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓)))
4117, 22, 40exlimd 1595 . . 3 ((𝜑𝑥 = 𝐴) → (∃𝑦 𝑦 = 𝐵 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓)))
4216, 41mpd 13 . 2 ((𝜑𝑥 = 𝐴) → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓))
431, 6, 11, 42exlimdd 1870 1 (𝜑 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wnf 1458  wex 1490  wcel 2146  wnfc 2304  Vcvv 2735  (class class class)co 5865  cmpo 5867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-setind 4530
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-iota 5170  df-fun 5210  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870
This theorem is referenced by:  ovmpodv  5997  ovmpodv2  5998
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