Theorem ralrnmptw 6864

Description: A restricted quantifier over an image set. Version of ralrnmpt 6866 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by Mario Carneiro, 20-Aug-2015.) (Revised by Gino Giotto, 26-Jan-2024.)

Hypotheses

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

Assertion

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

Distinct variable groups:   𝑥,𝑦   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 ralrnmptw

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

Step	Hyp	Ref	Expression
1		ralrnmptw.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
2	1	fnmpt 6471	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → 𝐹 Fn 𝐴)
3		dfsbcq 3681	. . . . 5 ⊢ (𝑤 = (𝐹‘𝑧) → ([𝑤 / 𝑦]𝜓 ↔ [(𝐹‘𝑧) / 𝑦]𝜓))
4	3	ralrn 6858	. . . 4 ⊢ (𝐹 Fn 𝐴 → (∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∀𝑧 ∈ 𝐴 [(𝐹‘𝑧) / 𝑦]𝜓))
5	2, 4	syl 17	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∀𝑧 ∈ 𝐴 [(𝐹‘𝑧) / 𝑦]𝜓))
6		nfsbc1v 3699	. . . 4 ⊢ Ⅎ𝑦[𝑤 / 𝑦]𝜓
7		nfv 1920	. . . 4 ⊢ Ⅎ𝑤𝜓
8		sbceq2a 3691	. . . 4 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑦]𝜓 ↔ 𝜓))
9	6, 7, 8	cbvralw 3339	. . 3 ⊢ (∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∀𝑦 ∈ ran 𝐹𝜓)
10		nfmpt1 5125	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
11	1, 10	nfcxfr 2897	. . . . . 6 ⊢ Ⅎ𝑥𝐹
12		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑥𝑧
13	11, 12	nffv 6678	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑧)
14		nfv 1920	. . . . 5 ⊢ Ⅎ𝑥𝜓
15	13, 14	nfsbcw 3701	. . . 4 ⊢ Ⅎ𝑥[(𝐹‘𝑧) / 𝑦]𝜓
16		nfv 1920	. . . 4 ⊢ Ⅎ𝑧[(𝐹‘𝑥) / 𝑦]𝜓
17		fveq2 6668	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
18	17	sbceq1d 3684	. . . 4 ⊢ (𝑧 = 𝑥 → ([(𝐹‘𝑧) / 𝑦]𝜓 ↔ [(𝐹‘𝑥) / 𝑦]𝜓))
19	15, 16, 18	cbvralw 3339	. . 3 ⊢ (∀𝑧 ∈ 𝐴 [(𝐹‘𝑧) / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓)
20	5, 9, 19	3bitr3g 316	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓))
21	1	fvmpt2 6780	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → (𝐹‘𝑥) = 𝐵)
22	21	sbceq1d 3684	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ [𝐵 / 𝑦]𝜓))
23		ralrnmptw.2	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
24	23	sbcieg 3718	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → ([𝐵 / 𝑦]𝜓 ↔ 𝜒))
25	24	adantl 485	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([𝐵 / 𝑦]𝜓 ↔ 𝜒))
26	22, 25	bitrd 282	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒))
27	26	ralimiaa 3074	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒))
28		ralbi 3082	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒) → (∀𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))
29	27, 28	syl 17	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))
30	20, 29	bitrd 282	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1542   ∈ wcel 2113  ∀wral 3053  [wsbc 3679   ↦ cmpt 5107  ran crn 5520   Fn wfn 6328  ‘cfv 6333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-fv 6341

This theorem is referenced by:  rexrnmptw  6865  ac6num  9972  gsumwspan  18120  dfod2  18802  ordtbaslem  21932  ordtrest2lem  21947  cncmp  22136  comppfsc  22276  ptpjopn  22356  ordthmeolem  22545  tsmsfbas  22872  tsmsf1o  22889  prdsxmetlem  23114  prdsbl  23237  metdsf  23593  metdsge  23594  minveclem1  24169  minveclem3b  24173  minveclem6  24179  mbflimsup  24411  xrlimcnp  25698  minvecolem1  28801  minvecolem5  28808  minvecolem6  28809  ordtrest2NEWlem  31436  cvmsss2  32799  fin2so  35376  prdsbnd  35563  rrnequiv  35605  ralrnmpt3  42326