Theorem imaeqalov 7689

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) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
imaeqalov	⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 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

Step	Hyp	Ref	Expression
1		df-ral 3068	. . 3 ⊢ (∀𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∀𝑥(𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) → 𝜑))
2		ovelimab 7628	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧)))
3	2	imbi1d 341	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → ((𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) → 𝜑) ↔ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑)))
4	3	albidv 1919	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∀𝑥(𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) → 𝜑) ↔ ∀𝑥(∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑)))
5	1, 4	bitrid 283	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∀𝑥(∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑)))
6		ralcom4 3292	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑥∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑥∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑))
7		r19.23v 3189	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ (∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑))
8	7	ralbii 3099	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑦 ∈ 𝐵 (∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑))
9		r19.23v 3189	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 (∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑))
10	8, 9	bitri 275	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑))
11	10	albii 1817	. . . 4 ⊢ (∀𝑥∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑥(∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑))
12	6, 11	bitri 275	. . 3 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑥∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑥(∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑))
13		ralcom4 3292	. . . . 5 ⊢ (∀𝑧 ∈ 𝐶 ∀𝑥(𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑥∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑))
14		ovex 7481	. . . . . . 7 ⊢ (𝑦𝐹𝑧) ∈ V
15		imaeqexov.1	. . . . . . 7 ⊢ (𝑥 = (𝑦𝐹𝑧) → (𝜑 ↔ 𝜓))
16	14, 15	ceqsalv 3529	. . . . . 6 ⊢ (∀𝑥(𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ 𝜓)
17	16	ralbii 3099	. . . . 5 ⊢ (∀𝑧 ∈ 𝐶 ∀𝑥(𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑧 ∈ 𝐶 𝜓)
18	13, 17	bitr3i 277	. . . 4 ⊢ (∀𝑥∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑧 ∈ 𝐶 𝜓)
19	18	ralbii 3099	. . 3 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑥∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓)
20	12, 19	bitr3i 277	. 2 ⊢ (∀𝑥(∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓)
21	5, 20	bitrdi 287	1 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ⊆ wss 3976   × cxp 5698   “ cima 5703   Fn wfn 6568  (class class class)co 7448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581  df-ov 7451

This theorem is referenced by:  naddunif  8749  naddasslem1  8750  naddasslem2  8751