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Mirrors > Home > MPE Home > Th. List > ralrnmptw | Structured version Visualization version GIF version |
Description: A restricted quantifier over an image set. Version of ralrnmpt 7098 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by Mario Carneiro, 20-Aug-2015.) Avoid ax-13 2370. (Revised by Gino Giotto, 26-Jan-2024.) |
Ref | Expression |
---|---|
ralrnmptw.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
ralrnmptw.2 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
ralrnmptw | ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralrnmptw.1 | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | 1 | fnmpt 6691 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → 𝐹 Fn 𝐴) |
3 | dfsbcq 3780 | . . . . 5 ⊢ (𝑤 = (𝐹‘𝑧) → ([𝑤 / 𝑦]𝜓 ↔ [(𝐹‘𝑧) / 𝑦]𝜓)) | |
4 | 3 | ralrn 7090 | . . . 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 3798 | . . . 4 ⊢ Ⅎ𝑦[𝑤 / 𝑦]𝜓 | |
7 | nfv 1916 | . . . 4 ⊢ Ⅎ𝑤𝜓 | |
8 | sbceq2a 3790 | . . . 4 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑦]𝜓 ↔ 𝜓)) | |
9 | 6, 7, 8 | cbvralw 3302 | . . 3 ⊢ (∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∀𝑦 ∈ ran 𝐹𝜓) |
10 | nfmpt1 5257 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | |
11 | 1, 10 | nfcxfr 2900 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 |
12 | nfcv 2902 | . . . . . 6 ⊢ Ⅎ𝑥𝑧 | |
13 | 11, 12 | nffv 6902 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
14 | nfv 1916 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
15 | 13, 14 | nfsbcw 3800 | . . . 4 ⊢ Ⅎ𝑥[(𝐹‘𝑧) / 𝑦]𝜓 |
16 | nfv 1916 | . . . 4 ⊢ Ⅎ𝑧[(𝐹‘𝑥) / 𝑦]𝜓 | |
17 | fveq2 6892 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) | |
18 | 17 | sbceq1d 3783 | . . . 4 ⊢ (𝑧 = 𝑥 → ([(𝐹‘𝑧) / 𝑦]𝜓 ↔ [(𝐹‘𝑥) / 𝑦]𝜓)) |
19 | 15, 16, 18 | cbvralw 3302 | . . 3 ⊢ (∀𝑧 ∈ 𝐴 [(𝐹‘𝑧) / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓) |
20 | 5, 9, 19 | 3bitr3g 312 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓)) |
21 | 1 | fvmpt2 7010 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → (𝐹‘𝑥) = 𝐵) |
22 | 21 | sbceq1d 3783 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ [𝐵 / 𝑦]𝜓)) |
23 | ralrnmptw.2 | . . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
24 | 23 | sbcieg 3818 | . . . . . 6 ⊢ (𝐵 ∈ 𝑉 → ([𝐵 / 𝑦]𝜓 ↔ 𝜒)) |
25 | 24 | adantl 481 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([𝐵 / 𝑦]𝜓 ↔ 𝜒)) |
26 | 22, 25 | bitrd 278 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒)) |
27 | 26 | ralimiaa 3081 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒)) |
28 | ralbi 3102 | . . 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 278 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 [wsbc 3778 ↦ cmpt 5232 ran crn 5678 Fn wfn 6539 ‘cfv 6544 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-fv 6552 |
This theorem is referenced by: rexrnmptw 7097 ac6num 10477 gsumwspan 18764 dfod2 19474 ordtbaslem 22913 ordtrest2lem 22928 cncmp 23117 comppfsc 23257 ptpjopn 23337 ordthmeolem 23526 tsmsfbas 23853 tsmsf1o 23870 prdsxmetlem 24095 prdsbl 24221 metdsf 24585 metdsge 24586 minveclem1 25173 minveclem3b 25177 minveclem6 25183 mbflimsup 25416 xrlimcnp 26706 minvecolem1 30391 minvecolem5 30398 minvecolem6 30399 ordtrest2NEWlem 33197 cvmsss2 34560 fin2so 36779 prdsbnd 36965 rrnequiv 37007 ralrnmpt3 44263 |
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