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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 7115 with a disjoint variable condition, which does not require ax-13 2376. (Contributed by Mario Carneiro, 20-Aug-2015.) Avoid ax-13 2376. (Revised by GG, 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 6707 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → 𝐹 Fn 𝐴) | 
| 3 | dfsbcq 3789 | . . . . 5 ⊢ (𝑤 = (𝐹‘𝑧) → ([𝑤 / 𝑦]𝜓 ↔ [(𝐹‘𝑧) / 𝑦]𝜓)) | |
| 4 | 3 | ralrn 7107 | . . . 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 3807 | . . . 4 ⊢ Ⅎ𝑦[𝑤 / 𝑦]𝜓 | |
| 7 | nfv 1913 | . . . 4 ⊢ Ⅎ𝑤𝜓 | |
| 8 | sbceq2a 3799 | . . . 4 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑦]𝜓 ↔ 𝜓)) | |
| 9 | 6, 7, 8 | cbvralw 3305 | . . 3 ⊢ (∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∀𝑦 ∈ ran 𝐹𝜓) | 
| 10 | nfmpt1 5249 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 11 | 1, 10 | nfcxfr 2902 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | 
| 12 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑥𝑧 | |
| 13 | 11, 12 | nffv 6915 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑧) | 
| 14 | nfv 1913 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
| 15 | 13, 14 | nfsbcw 3809 | . . . 4 ⊢ Ⅎ𝑥[(𝐹‘𝑧) / 𝑦]𝜓 | 
| 16 | nfv 1913 | . . . 4 ⊢ Ⅎ𝑧[(𝐹‘𝑥) / 𝑦]𝜓 | |
| 17 | fveq2 6905 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) | |
| 18 | 17 | sbceq1d 3792 | . . . 4 ⊢ (𝑧 = 𝑥 → ([(𝐹‘𝑧) / 𝑦]𝜓 ↔ [(𝐹‘𝑥) / 𝑦]𝜓)) | 
| 19 | 15, 16, 18 | cbvralw 3305 | . . 3 ⊢ (∀𝑧 ∈ 𝐴 [(𝐹‘𝑧) / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓) | 
| 20 | 5, 9, 19 | 3bitr3g 313 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓)) | 
| 21 | 1 | fvmpt2 7026 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → (𝐹‘𝑥) = 𝐵) | 
| 22 | 21 | sbceq1d 3792 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ [𝐵 / 𝑦]𝜓)) | 
| 23 | ralrnmptw.2 | . . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
| 24 | 23 | sbcieg 3827 | . . . . . 6 ⊢ (𝐵 ∈ 𝑉 → ([𝐵 / 𝑦]𝜓 ↔ 𝜒)) | 
| 25 | 24 | adantl 481 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([𝐵 / 𝑦]𝜓 ↔ 𝜒)) | 
| 26 | 22, 25 | bitrd 279 | . . . 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 279 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 [wsbc 3787 ↦ cmpt 5224 ran crn 5685 Fn wfn 6555 ‘cfv 6560 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-fv 6568 | 
| This theorem is referenced by: rexrnmptw 7114 ac6num 10520 gsumwspan 18860 dfod2 19583 ordtbaslem 23197 ordtrest2lem 23212 cncmp 23401 comppfsc 23541 ptpjopn 23621 ordthmeolem 23810 tsmsfbas 24137 tsmsf1o 24154 prdsxmetlem 24379 prdsbl 24505 metdsf 24871 metdsge 24872 minveclem1 25459 minveclem3b 25463 minveclem6 25469 mbflimsup 25702 xrlimcnp 27012 minvecolem1 30894 minvecolem5 30901 minvecolem6 30902 ordtrest2NEWlem 33922 cvmsss2 35280 fin2so 37615 prdsbnd 37801 rrnequiv 37843 ralrnmpt3 45271 | 
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