Theorem rexrnmpt 5661

Description: A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)

Hypotheses

Ref	Expression
ralrnmpt.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
ralrnmpt.2	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
rexrnmpt	⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))

Distinct variable groups:   𝑥,𝐴   𝑦,𝐵   𝜒,𝑦   𝑦,𝐹   𝜓,𝑥

Allowed substitution hints:   𝜓(𝑦)   𝜒(𝑥)   𝐴(𝑦)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem rexrnmpt

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ralrnmpt.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
2	1	fnmpt 5344	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → 𝐹 Fn 𝐴)
3		dfsbcq 2966	. . . . 5 ⊢ (𝑤 = (𝐹‘𝑧) → ([𝑤 / 𝑦]𝜓 ↔ [(𝐹‘𝑧) / 𝑦]𝜓))
4	3	rexrn 5655	. . . 4 ⊢ (𝐹 Fn 𝐴 → (∃𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∃𝑧 ∈ 𝐴 [(𝐹‘𝑧) / 𝑦]𝜓))
5	2, 4	syl 14	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∃𝑧 ∈ 𝐴 [(𝐹‘𝑧) / 𝑦]𝜓))
6		nfv 1528	. . . . 5 ⊢ Ⅎ𝑤𝜓
7		nfsbc1v 2983	. . . . 5 ⊢ Ⅎ𝑦[𝑤 / 𝑦]𝜓
8		sbceq1a 2974	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝜓 ↔ [𝑤 / 𝑦]𝜓))
9	6, 7, 8	cbvrex 2702	. . . 4 ⊢ (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓)
10	9	bicomi 132	. . 3 ⊢ (∃𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∃𝑦 ∈ ran 𝐹𝜓)
11		nfmpt1 4098	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
12	1, 11	nfcxfr 2316	. . . . . 6 ⊢ Ⅎ𝑥𝐹
13		nfcv 2319	. . . . . 6 ⊢ Ⅎ𝑥𝑧
14	12, 13	nffv 5527	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑧)
15		nfv 1528	. . . . 5 ⊢ Ⅎ𝑥𝜓
16	14, 15	nfsbc 2985	. . . 4 ⊢ Ⅎ𝑥[(𝐹‘𝑧) / 𝑦]𝜓
17		nfv 1528	. . . 4 ⊢ Ⅎ𝑧[(𝐹‘𝑥) / 𝑦]𝜓
18		fveq2 5517	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
19	18	sbceq1d 2969	. . . 4 ⊢ (𝑧 = 𝑥 → ([(𝐹‘𝑧) / 𝑦]𝜓 ↔ [(𝐹‘𝑥) / 𝑦]𝜓))
20	16, 17, 19	cbvrex 2702	. . 3 ⊢ (∃𝑧 ∈ 𝐴 [(𝐹‘𝑧) / 𝑦]𝜓 ↔ ∃𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓)
21	5, 10, 20	3bitr3g 222	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓))
22	1	fvmpt2 5601	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → (𝐹‘𝑥) = 𝐵)
23	22	sbceq1d 2969	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ [𝐵 / 𝑦]𝜓))
24		ralrnmpt.2	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
25	24	sbcieg 2997	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → ([𝐵 / 𝑦]𝜓 ↔ 𝜒))
26	25	adantl 277	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([𝐵 / 𝑦]𝜓 ↔ 𝜒))
27	23, 26	bitrd 188	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒))
28	27	ralimiaa 2539	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒))
29		pm5.32 453	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒)) ↔ ((𝑥 ∈ 𝐴 ∧ [(𝐹‘𝑥) / 𝑦]𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
30	29	albii 1470	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒)) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ [(𝐹‘𝑥) / 𝑦]𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
31		exbi 1604	. . . . 5 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ [(𝐹‘𝑥) / 𝑦]𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)) → (∃𝑥(𝑥 ∈ 𝐴 ∧ [(𝐹‘𝑥) / 𝑦]𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)))
32	30, 31	sylbi 121	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒)) → (∃𝑥(𝑥 ∈ 𝐴 ∧ [(𝐹‘𝑥) / 𝑦]𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)))
33		df-ral 2460	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒)))
34		df-rex 2461	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ [(𝐹‘𝑥) / 𝑦]𝜓))
35		df-rex 2461	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝜒 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜒))
36	34, 35	bibi12i 229	. . . 4 ⊢ ((∃𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ [(𝐹‘𝑥) / 𝑦]𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)))
37	32, 33, 36	3imtr4i 201	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒) → (∃𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))
38	28, 37	syl 14	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))
39	21, 38	bitrd 188	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1351   = wceq 1353  ∃wex 1492   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456  [wsbc 2964   ↦ cmpt 4066  ran crn 4629   Fn wfn 5213  ‘cfv 5218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226

This theorem is referenced by:  txbas  13797