Theorem imaeqalov 7600

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 3053	. . 3 ⊢ (∀𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∀𝑥(𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) → 𝜑))
2		ovelimab 7539	. . . . 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 1922	. . 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 3264	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑥∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑥∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑))
7		r19.23v 3165	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ (∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑))
8	7	ralbii 3084	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑦 ∈ 𝐵 (∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑))
9		r19.23v 3165	. . . . . 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 1821	. . . 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 3264	. . . . 5 ⊢ (∀𝑧 ∈ 𝐶 ∀𝑥(𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑥∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑))
14		ovex 7394	. . . . . . 7 ⊢ (𝑦𝐹𝑧) ∈ V
15		imaeqexov.1	. . . . . . 7 ⊢ (𝑥 = (𝑦𝐹𝑧) → (𝜑 ↔ 𝜓))
16	14, 15	ceqsalv 3470	. . . . . 6 ⊢ (∀𝑥(𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ 𝜓)
17	16	ralbii 3084	. . . . 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 3084	. . 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 1540   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ⊆ wss 3890   × cxp 5623   “ cima 5628   Fn wfn 6488  (class class class)co 7361

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-fv 6501  df-ov 7364

This theorem is referenced by:  naddunif  8623  naddasslem1  8624  naddasslem2  8625