Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > elfvmptrab1 | 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. (Contributed by Alexander van der Vekens, 15-Jul-2018.) |
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 | elfvmptrab1.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑}) | |
2 | 1 | funmpt2 5237 | . . . 4 ⊢ Fun 𝐹 |
3 | funrel 5215 | . . . 4 ⊢ (Fun 𝐹 → Rel 𝐹) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ Rel 𝐹 |
5 | relelfvdm 5528 | . . 3 ⊢ ((Rel 𝐹 ∧ 𝑌 ∈ (𝐹‘𝑋)) → 𝑋 ∈ dom 𝐹) | |
6 | 4, 5 | mpan 422 | . 2 ⊢ (𝑌 ∈ (𝐹‘𝑋) → 𝑋 ∈ dom 𝐹) |
7 | 1 | dmmptss 5107 | . . . . . 6 ⊢ dom 𝐹 ⊆ 𝑉 |
8 | 7 | sseli 3143 | . . . . 5 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ 𝑉) |
9 | elfvmptrab1.v | . . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 ∈ V) | |
10 | rabexg 4132 | . . . . . 6 ⊢ (⦋𝑋 / 𝑚⦌𝑀 ∈ V → {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} ∈ V) | |
11 | 8, 9, 10 | 3syl 17 | . . . . 5 ⊢ (𝑋 ∈ dom 𝐹 → {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} ∈ V) |
12 | nfcv 2312 | . . . . . 6 ⊢ Ⅎ𝑥𝑋 | |
13 | nfsbc1v 2973 | . . . . . . 7 ⊢ Ⅎ𝑥[𝑋 / 𝑥]𝜑 | |
14 | nfcv 2312 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑀 | |
15 | 12, 14 | nfcsb 3086 | . . . . . . 7 ⊢ Ⅎ𝑥⦋𝑋 / 𝑚⦌𝑀 |
16 | 13, 15 | nfrabxy 2650 | . . . . . 6 ⊢ Ⅎ𝑥{𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} |
17 | csbeq1 3052 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → ⦋𝑥 / 𝑚⦌𝑀 = ⦋𝑋 / 𝑚⦌𝑀) | |
18 | sbceq1a 2964 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝜑 ↔ [𝑋 / 𝑥]𝜑)) | |
19 | 17, 18 | rabeqbidv 2725 | . . . . . 6 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑} = {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑}) |
20 | 12, 16, 19, 1 | fvmptf 5588 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} ∈ V) → (𝐹‘𝑋) = {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑}) |
21 | 8, 11, 20 | syl2anc 409 | . . . 4 ⊢ (𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑}) |
22 | 21 | eleq2d 2240 | . . 3 ⊢ (𝑋 ∈ dom 𝐹 → (𝑌 ∈ (𝐹‘𝑋) ↔ 𝑌 ∈ {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑})) |
23 | elrabi 2883 | . . . . 5 ⊢ (𝑌 ∈ {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} → 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀) | |
24 | 8, 23 | anim12i 336 | . . . 4 ⊢ ((𝑋 ∈ dom 𝐹 ∧ 𝑌 ∈ {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑}) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀)) |
25 | 24 | ex 114 | . . 3 ⊢ (𝑋 ∈ dom 𝐹 → (𝑌 ∈ {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀))) |
26 | 22, 25 | sylbid 149 | . 2 ⊢ (𝑋 ∈ dom 𝐹 → (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀))) |
27 | 6, 26 | mpcom 36 | 1 ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 {crab 2452 Vcvv 2730 [wsbc 2955 ⦋csb 3049 ↦ cmpt 4050 dom cdm 4611 Rel wrel 4616 Fun wfun 5192 ‘cfv 5198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fv 5206 |
This theorem is referenced by: elfvmptrab 5591 |
Copyright terms: Public domain | W3C validator |