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Mirrors > Home > MPE Home > Th. List > elfvmptrab1 | Structured version Visualization version GIF version |
Description: Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. Here, the base set of the class abstraction depends on the argument of the function. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker elfvmptrab1w 6796 when possible. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elfvmptrab1.f | ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑}) |
elfvmptrab1.v | ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 ∈ V) |
Ref | Expression |
---|---|
elfvmptrab1 | ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ne0i 4302 | . . 3 ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝐹‘𝑋) ≠ ∅) | |
2 | ndmfv 6702 | . . . 4 ⊢ (¬ 𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = ∅) | |
3 | 2 | necon1ai 3045 | . . 3 ⊢ ((𝐹‘𝑋) ≠ ∅ → 𝑋 ∈ dom 𝐹) |
4 | elfvmptrab1.f | . . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑}) | |
5 | 4 | dmmptss 6097 | . . . . . . 7 ⊢ dom 𝐹 ⊆ 𝑉 |
6 | 5 | sseli 3965 | . . . . . 6 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ 𝑉) |
7 | elfvmptrab1.v | . . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 ∈ V) | |
8 | rabexg 5236 | . . . . . . 7 ⊢ (⦋𝑋 / 𝑚⦌𝑀 ∈ V → {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} ∈ V) | |
9 | 6, 7, 8 | 3syl 18 | . . . . . 6 ⊢ (𝑋 ∈ dom 𝐹 → {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} ∈ V) |
10 | nfcv 2979 | . . . . . . 7 ⊢ Ⅎ𝑥𝑋 | |
11 | nfsbc1v 3794 | . . . . . . . 8 ⊢ Ⅎ𝑥[𝑋 / 𝑥]𝜑 | |
12 | nfcv 2979 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝑀 | |
13 | 10, 12 | nfcsb 3912 | . . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑋 / 𝑚⦌𝑀 |
14 | 11, 13 | nfrab 3388 | . . . . . . 7 ⊢ Ⅎ𝑥{𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} |
15 | csbeq1 3888 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → ⦋𝑥 / 𝑚⦌𝑀 = ⦋𝑋 / 𝑚⦌𝑀) | |
16 | sbceq1a 3785 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝜑 ↔ [𝑋 / 𝑥]𝜑)) | |
17 | 15, 16 | rabeqbidv 3487 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑} = {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑}) |
18 | 10, 14, 17, 4 | fvmptf 6791 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} ∈ V) → (𝐹‘𝑋) = {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑}) |
19 | 6, 9, 18 | syl2anc 586 | . . . . 5 ⊢ (𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑}) |
20 | 19 | eleq2d 2900 | . . . 4 ⊢ (𝑋 ∈ dom 𝐹 → (𝑌 ∈ (𝐹‘𝑋) ↔ 𝑌 ∈ {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑})) |
21 | elrabi 3677 | . . . . . 6 ⊢ (𝑌 ∈ {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} → 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀) | |
22 | 6, 21 | anim12i 614 | . . . . 5 ⊢ ((𝑋 ∈ dom 𝐹 ∧ 𝑌 ∈ {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑}) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀)) |
23 | 22 | ex 415 | . . . 4 ⊢ (𝑋 ∈ dom 𝐹 → (𝑌 ∈ {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀))) |
24 | 20, 23 | sylbid 242 | . . 3 ⊢ (𝑋 ∈ dom 𝐹 → (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀))) |
25 | 1, 3, 24 | 3syl 18 | . 2 ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀))) |
26 | 25 | pm2.43i 52 | 1 ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 {crab 3144 Vcvv 3496 [wsbc 3774 ⦋csb 3885 ∅c0 4293 ↦ cmpt 5148 dom cdm 5557 ‘cfv 6357 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-13 2390 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fv 6365 |
This theorem is referenced by: (None) |
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