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Theorem imaeqalov 7649
Description: Substitute an operation value into a universal quantifier over an image. (Contributed by Scott Fenton, 20-Jan-2025.)
Hypothesis
Ref Expression
imaeqexov.1 (𝑥 = (𝑦𝐹𝑧) → (𝜑𝜓))
Assertion
Ref Expression
imaeqalov ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∀𝑦𝐵𝑧𝐶 𝜓))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝜑,𝑦,𝑧   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦,𝑧)

Proof of Theorem imaeqalov
StepHypRef Expression
1 df-ral 3061 . . 3 (∀𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∀𝑥(𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) → 𝜑))
2 ovelimab 7588 . . . . 5 ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) ↔ ∃𝑦𝐵𝑧𝐶 𝑥 = (𝑦𝐹𝑧)))
32imbi1d 341 . . . 4 ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → ((𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) → 𝜑) ↔ (∃𝑦𝐵𝑧𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑)))
43albidv 1922 . . 3 ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∀𝑥(𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) → 𝜑) ↔ ∀𝑥(∃𝑦𝐵𝑧𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑)))
51, 4bitrid 283 . 2 ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∀𝑥(∃𝑦𝐵𝑧𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑)))
6 ralcom4 3282 . . . 4 (∀𝑦𝐵𝑥𝑧𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑥𝑦𝐵𝑧𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑))
7 r19.23v 3181 . . . . . . 7 (∀𝑧𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ (∃𝑧𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑))
87ralbii 3092 . . . . . 6 (∀𝑦𝐵𝑧𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑦𝐵 (∃𝑧𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑))
9 r19.23v 3181 . . . . . 6 (∀𝑦𝐵 (∃𝑧𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ (∃𝑦𝐵𝑧𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑))
108, 9bitri 275 . . . . 5 (∀𝑦𝐵𝑧𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ (∃𝑦𝐵𝑧𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑))
1110albii 1820 . . . 4 (∀𝑥𝑦𝐵𝑧𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑥(∃𝑦𝐵𝑧𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑))
126, 11bitri 275 . . 3 (∀𝑦𝐵𝑥𝑧𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑥(∃𝑦𝐵𝑧𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑))
13 ralcom4 3282 . . . . 5 (∀𝑧𝐶𝑥(𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑥𝑧𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑))
14 ovex 7445 . . . . . . 7 (𝑦𝐹𝑧) ∈ V
15 imaeqexov.1 . . . . . . 7 (𝑥 = (𝑦𝐹𝑧) → (𝜑𝜓))
1614, 15ceqsalv 3511 . . . . . 6 (∀𝑥(𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ 𝜓)
1716ralbii 3092 . . . . 5 (∀𝑧𝐶𝑥(𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑧𝐶 𝜓)
1813, 17bitr3i 277 . . . 4 (∀𝑥𝑧𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑧𝐶 𝜓)
1918ralbii 3092 . . 3 (∀𝑦𝐵𝑥𝑧𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑦𝐵𝑧𝐶 𝜓)
2012, 19bitr3i 277 . 2 (∀𝑥(∃𝑦𝐵𝑧𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑦𝐵𝑧𝐶 𝜓)
215, 20bitrdi 287 1 ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∀𝑦𝐵𝑧𝐶 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1538   = wceq 1540  wcel 2105  wral 3060  wrex 3069  wss 3948   × cxp 5674  cima 5679   Fn wfn 6538  (class class class)co 7412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  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-iun 4999  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-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551  df-ov 7415
This theorem is referenced by:  naddunif  8695  naddasslem1  8696  naddasslem2  8697
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