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Theorem ovmpodf 7566
Description: Alternate deduction version of ovmpo 7570, 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 1917 . 2 𝑥𝜑
2 ovmpodf.5 . . . 4 𝑥𝐹
3 nfmpo1 7491 . . . 4 𝑥(𝑥𝐶, 𝑦𝐷𝑅)
42, 3nfeq 2916 . . 3 𝑥 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
5 ovmpodf.6 . . 3 𝑥𝜓
64, 5nfim 1899 . 2 𝑥(𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓)
7 ovmpodf.1 . . . 4 (𝜑𝐴𝐶)
87elexd 3494 . . 3 (𝜑𝐴 ∈ V)
9 isset 3487 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
108, 9sylib 217 . 2 (𝜑 → ∃𝑥 𝑥 = 𝐴)
11 nfv 1917 . . 3 𝑦(𝜑𝑥 = 𝐴)
12 ovmpodf.7 . . . . 5 𝑦𝐹
13 nfmpo2 7492 . . . . 5 𝑦(𝑥𝐶, 𝑦𝐷𝑅)
1412, 13nfeq 2916 . . . 4 𝑦 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
15 ovmpodf.8 . . . 4 𝑦𝜓
1614, 15nfim 1899 . . 3 𝑦(𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓)
17 ovmpodf.2 . . . . 5 ((𝜑𝑥 = 𝐴) → 𝐵𝐷)
1817elexd 3494 . . . 4 ((𝜑𝑥 = 𝐴) → 𝐵 ∈ V)
19 isset 3487 . . . 4 (𝐵 ∈ V ↔ ∃𝑦 𝑦 = 𝐵)
2018, 19sylib 217 . . 3 ((𝜑𝑥 = 𝐴) → ∃𝑦 𝑦 = 𝐵)
21 oveq 7417 . . . . 5 (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → (𝐴𝐹𝐵) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵))
22 simprl 769 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑥 = 𝐴)
23 simprr 771 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑦 = 𝐵)
2422, 23oveq12d 7429 . . . . . . . 8 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵))
257adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝐴𝐶)
2622, 25eqeltrd 2833 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑥𝐶)
2717adantrr 715 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝐵𝐷)
2823, 27eqeltrd 2833 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑦𝐷)
29 ovmpodf.3 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅𝑉)
30 eqid 2732 . . . . . . . . . 10 (𝑥𝐶, 𝑦𝐷𝑅) = (𝑥𝐶, 𝑦𝐷𝑅)
3130ovmpt4g 7557 . . . . . . . . 9 ((𝑥𝐶𝑦𝐷𝑅𝑉) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)
3226, 28, 29, 31syl3anc 1371 . . . . . . . 8 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)
3324, 32eqtr3d 2774 . . . . . . 7 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑅)
3433eqeq2d 2743 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) ↔ (𝐴𝐹𝐵) = 𝑅))
35 ovmpodf.4 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = 𝑅𝜓))
3634, 35sylbid 239 . . . . 5 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) → 𝜓))
3721, 36syl5 34 . . . 4 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓))
3837expr 457 . . 3 ((𝜑𝑥 = 𝐴) → (𝑦 = 𝐵 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓)))
3911, 16, 20, 38exlimimdd 2212 . 2 ((𝜑𝑥 = 𝐴) → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓))
401, 6, 10, 39exlimdd 2213 1 (𝜑 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wex 1781  wnf 1785  wcel 2106  wnfc 2883  Vcvv 3474  (class class class)co 7411  cmpo 7413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416
This theorem is referenced by:  ovmpodv  7567  ovmpodv2  7568
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