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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 5208 | . . . 4 ⊢ Fun 𝐹 |
3 | funrel 5186 | . . . 4 ⊢ (Fun 𝐹 → Rel 𝐹) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ Rel 𝐹 |
5 | relelfvdm 5499 | . . 3 ⊢ ((Rel 𝐹 ∧ 𝑌 ∈ (𝐹‘𝑋)) → 𝑋 ∈ dom 𝐹) | |
6 | 4, 5 | mpan 421 | . 2 ⊢ (𝑌 ∈ (𝐹‘𝑋) → 𝑋 ∈ dom 𝐹) |
7 | 1 | dmmptss 5081 | . . . . . 6 ⊢ dom 𝐹 ⊆ 𝑉 |
8 | 7 | sseli 3124 | . . . . 5 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ 𝑉) |
9 | elfvmptrab1.v | . . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 ∈ V) | |
10 | rabexg 4107 | . . . . . 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 2299 | . . . . . 6 ⊢ Ⅎ𝑥𝑋 | |
13 | nfsbc1v 2955 | . . . . . . 7 ⊢ Ⅎ𝑥[𝑋 / 𝑥]𝜑 | |
14 | nfcv 2299 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑀 | |
15 | 12, 14 | nfcsb 3068 | . . . . . . 7 ⊢ Ⅎ𝑥⦋𝑋 / 𝑚⦌𝑀 |
16 | 13, 15 | nfrabxy 2637 | . . . . . 6 ⊢ Ⅎ𝑥{𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} |
17 | csbeq1 3034 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → ⦋𝑥 / 𝑚⦌𝑀 = ⦋𝑋 / 𝑚⦌𝑀) | |
18 | sbceq1a 2946 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝜑 ↔ [𝑋 / 𝑥]𝜑)) | |
19 | 17, 18 | rabeqbidv 2707 | . . . . . 6 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑} = {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑}) |
20 | 12, 16, 19, 1 | fvmptf 5559 | . . . . 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 2227 | . . 3 ⊢ (𝑋 ∈ dom 𝐹 → (𝑌 ∈ (𝐹‘𝑋) ↔ 𝑌 ∈ {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑})) |
23 | elrabi 2865 | . . . . 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 1335 ∈ wcel 2128 {crab 2439 Vcvv 2712 [wsbc 2937 ⦋csb 3031 ↦ cmpt 4025 dom cdm 4585 Rel wrel 4590 Fun wfun 5163 ‘cfv 5169 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fv 5177 |
This theorem is referenced by: elfvmptrab 5562 |
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