Theorem imaeqalov 7672

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 3062	. . 3 ⊢ (∀𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∀𝑥(𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) → 𝜑))
2		ovelimab 7611	. . . . 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 1920	. . 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 3286	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑥∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑥∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑))
7		r19.23v 3183	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ (∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑))
8	7	ralbii 3093	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑦 ∈ 𝐵 (∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑))
9		r19.23v 3183	. . . . . 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 1819	. . . 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 3286	. . . . 5 ⊢ (∀𝑧 ∈ 𝐶 ∀𝑥(𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑥∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑))
14		ovex 7464	. . . . . . 7 ⊢ (𝑦𝐹𝑧) ∈ V
15		imaeqexov.1	. . . . . . 7 ⊢ (𝑥 = (𝑦𝐹𝑧) → (𝜑 ↔ 𝜓))
16	14, 15	ceqsalv 3521	. . . . . 6 ⊢ (∀𝑥(𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ 𝜓)
17	16	ralbii 3093	. . . . 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 3093	. . 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 1538   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070   ⊆ wss 3951   × cxp 5683   “ cima 5688   Fn wfn 6556  (class class class)co 7431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-ov 7434

This theorem is referenced by:  naddunif  8731  naddasslem1  8732  naddasslem2  8733