Theorem ralrnmpo 7485

Description: A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)

Hypotheses

Ref	Expression
rngop.1	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
ralrnmpo.2	⊢ (𝑧 = 𝐶 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ralrnmpo	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓))

Distinct variable groups:   𝑦,𝑧,𝐴   𝑧,𝐵   𝑧,𝐶   𝑧,𝐹   𝜓,𝑧   𝑥,𝑦,𝑧   𝜑,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑧)   𝜓(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem ralrnmpo

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rngop.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
2	1	rnmpo 7479	. . . 4 ⊢ ran 𝐹 = {𝑤 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶}
3	2	raleqi 3290	. . 3 ⊢ (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑧 ∈ {𝑤 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶}𝜑)
4		eqeq1 2735	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 = 𝐶 ↔ 𝑧 = 𝐶))
5	4	2rexbidv 3197	. . . 4 ⊢ (𝑤 = 𝑧 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶))
6	5	ralab 3647	. . 3 ⊢ (∀𝑧 ∈ {𝑤 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶}𝜑 ↔ ∀𝑧(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
7		ralcom4 3258	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑧∀𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
8		r19.23v 3159	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
9	8	albii 1820	. . . 4 ⊢ (∀𝑧∀𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑧(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
10	7, 9	bitr2i 276	. . 3 ⊢ (∀𝑧(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
11	3, 6, 10	3bitri 297	. 2 ⊢ (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
12		ralcom4 3258	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ ∀𝑧∀𝑦 ∈ 𝐵 (𝑧 = 𝐶 → 𝜑))
13		r19.23v 3159	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐵 (𝑧 = 𝐶 → 𝜑) ↔ (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
14	13	albii 1820	. . . . . 6 ⊢ (∀𝑧∀𝑦 ∈ 𝐵 (𝑧 = 𝐶 → 𝜑) ↔ ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
15	12, 14	bitri 275	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
16		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑧𝜓
17		ralrnmpo.2	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝜑 ↔ 𝜓))
18	16, 17	ceqsalg 3472	. . . . . . 7 ⊢ (𝐶 ∈ 𝑉 → (∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ 𝜓))
19	18	ralimi 3069	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ∀𝑦 ∈ 𝐵 (∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ 𝜓))
20		ralbi 3087	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 (∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ 𝜓) → (∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ ∀𝑦 ∈ 𝐵 𝜓))
21	19, 20	syl 17	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ ∀𝑦 ∈ 𝐵 𝜓))
22	15, 21	bitr3id 285	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑦 ∈ 𝐵 𝜓))
23	22	ralimi 3069	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 (∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑦 ∈ 𝐵 𝜓))
24		ralbi 3087	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑦 ∈ 𝐵 𝜓) → (∀𝑥 ∈ 𝐴 ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓))
25	23, 24	syl 17	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓))
26	11, 25	bitrid 283	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1539   = wceq 1541   ∈ wcel 2111  {cab 2709  ∀wral 3047  ∃wrex 3056  ran crn 5615   ∈ cmpo 7348

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-cnv 5622  df-dm 5624  df-rn 5625  df-oprab 7350  df-mpo 7351

This theorem is referenced by:  rexrnmpo  7486  efgval2  19636  txcnp  23535  txcnmpt  23539  txflf  23921