Theorem imaeqalov 7654

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 3077	. . 3 ⊢ (∀𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∀𝑥(𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) → 𝜑))
2		ovelimab 7593	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧)))
3	2	imbi1d 344	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → ((𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) → 𝜑) ↔ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑)))
4	3	albidv 1953	. . 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 286	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∀𝑥(∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑)))
6		ralcom4 3288	. . . 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 3108	. . . . . 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 278	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑))
11	10	albii 1852	. . . 4 ⊢ (∀𝑥∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑥(∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑))
12	6, 11	bitri 278	. . 3 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑥∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑥(∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑))
13		ralcom4 3288	. . . . 5 ⊢ (∀𝑧 ∈ 𝐶 ∀𝑥(𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑥∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑))
14		ovex 7447	. . . . . . 7 ⊢ (𝑦𝐹𝑧) ∈ V
15		imaeqexov.1	. . . . . . 7 ⊢ (𝑥 = (𝑦𝐹𝑧) → (𝜑 ↔ 𝜓))
16	14, 15	ceqsalv 3489	. . . . . 6 ⊢ (∀𝑥(𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ 𝜓)
17	16	ralbii 3108	. . . . 5 ⊢ (∀𝑧 ∈ 𝐶 ∀𝑥(𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑧 ∈ 𝐶 𝜓)
18	13, 17	bitr3i 280	. . . 4 ⊢ (∀𝑥∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑧 ∈ 𝐶 𝜓)
19	18	ralbii 3108	. . 3 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑥∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓)
20	12, 19	bitr3i 280	. 2 ⊢ (∀𝑥(∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓)
21	5, 20	bitrdi 290	1 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   ⊆ wss 3899   × cxp 5653   “ cima 5658   Fn wfn 6528  (class class class)co 7414

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-fv 6541  df-ov 7417

This theorem is used by:  naddunif  8683  naddasslem1  8684  naddasslem2  8685