Theorem rexrnmpo 5968

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
rexrnmpo	⊢ (∀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 rexrnmpo

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 5963	. . . 4 ⊢ ran 𝐹 = {𝑤 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶}
3	2	rexeqi 2670	. . 3 ⊢ (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑧 ∈ {𝑤 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶}𝜑)
4		eqeq1 2177	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 = 𝐶 ↔ 𝑧 = 𝐶))
5	4	2rexbidv 2495	. . . 4 ⊢ (𝑤 = 𝑧 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶))
6	5	rexab 2892	. . 3 ⊢ (∃𝑧 ∈ {𝑤 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶}𝜑 ↔ ∃𝑧(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
7		rexcom4 2753	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
8		r19.41v 2626	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
9	8	exbii 1598	. . . 4 ⊢ (∃𝑧∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑧(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
10	7, 9	bitr2i 184	. . 3 ⊢ (∃𝑧(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐴 ∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
11	3, 6, 10	3bitri 205	. 2 ⊢ (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
12		rexcom4 2753	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧(𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑧∃𝑦 ∈ 𝐵 (𝑧 = 𝐶 ∧ 𝜑))
13		r19.41v 2626	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐵 (𝑧 = 𝐶 ∧ 𝜑) ↔ (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
14	13	exbii 1598	. . . . . 6 ⊢ (∃𝑧∃𝑦 ∈ 𝐵 (𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
15	12, 14	bitri 183	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧(𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
16		ralrnmpo.2	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝜑 ↔ 𝜓))
17	16	ceqsexgv 2859	. . . . . . 7 ⊢ (𝐶 ∈ 𝑉 → (∃𝑧(𝑧 = 𝐶 ∧ 𝜑) ↔ 𝜓))
18	17	ralimi 2533	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ∀𝑦 ∈ 𝐵 (∃𝑧(𝑧 = 𝐶 ∧ 𝜑) ↔ 𝜓))
19		rexbi 2603	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 (∃𝑧(𝑧 = 𝐶 ∧ 𝜑) ↔ 𝜓) → (∃𝑦 ∈ 𝐵 ∃𝑧(𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 𝜓))
20	18, 19	syl 14	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∃𝑦 ∈ 𝐵 ∃𝑧(𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 𝜓))
21	15, 20	bitr3id 193	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 𝜓))
22	21	ralimi 2533	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 (∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 𝜓))
23		rexbi 2603	. . 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))
24	22, 23	syl 14	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∃𝑥 ∈ 𝐴 ∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓))
25	11, 24	syl5bb 191	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348  ∃wex 1485   ∈ wcel 2141  {cab 2156  ∀wral 2448  ∃wrex 2449  ran crn 4612   ∈ cmpo 5855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-cnv 4619  df-dm 4621  df-rn 4622  df-oprab 5857  df-mpo 5858

This theorem is referenced by:  eltx  13053  txrest  13070  txlm  13073