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Mirrors > Home > MPE Home > Th. List > resfunexgALT | Structured version Visualization version GIF version |
Description: Alternate proof of resfunexg 6955, shorter but requiring ax-pow 5231 and ax-un 7441. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
resfunexgALT | ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmresexg 5842 | . . . 4 ⊢ (𝐵 ∈ 𝐶 → dom (𝐴 ↾ 𝐵) ∈ V) | |
2 | 1 | adantl 485 | . . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → dom (𝐴 ↾ 𝐵) ∈ V) |
3 | df-ima 5532 | . . . 4 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
4 | funimaexg 6410 | . . . 4 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V) | |
5 | 3, 4 | eqeltrrid 2895 | . . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → ran (𝐴 ↾ 𝐵) ∈ V) |
6 | 2, 5 | jca 515 | . 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (dom (𝐴 ↾ 𝐵) ∈ V ∧ ran (𝐴 ↾ 𝐵) ∈ V)) |
7 | xpexg 7453 | . 2 ⊢ ((dom (𝐴 ↾ 𝐵) ∈ V ∧ ran (𝐴 ↾ 𝐵) ∈ V) → (dom (𝐴 ↾ 𝐵) × ran (𝐴 ↾ 𝐵)) ∈ V) | |
8 | relres 5847 | . . . 4 ⊢ Rel (𝐴 ↾ 𝐵) | |
9 | relssdmrn 6088 | . . . 4 ⊢ (Rel (𝐴 ↾ 𝐵) → (𝐴 ↾ 𝐵) ⊆ (dom (𝐴 ↾ 𝐵) × ran (𝐴 ↾ 𝐵))) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ (𝐴 ↾ 𝐵) ⊆ (dom (𝐴 ↾ 𝐵) × ran (𝐴 ↾ 𝐵)) |
11 | ssexg 5191 | . . 3 ⊢ (((𝐴 ↾ 𝐵) ⊆ (dom (𝐴 ↾ 𝐵) × ran (𝐴 ↾ 𝐵)) ∧ (dom (𝐴 ↾ 𝐵) × ran (𝐴 ↾ 𝐵)) ∈ V) → (𝐴 ↾ 𝐵) ∈ V) | |
12 | 10, 11 | mpan 689 | . 2 ⊢ ((dom (𝐴 ↾ 𝐵) × ran (𝐴 ↾ 𝐵)) ∈ V → (𝐴 ↾ 𝐵) ∈ V) |
13 | 6, 7, 12 | 3syl 18 | 1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 × cxp 5517 dom cdm 5519 ran crn 5520 ↾ cres 5521 “ cima 5522 Rel wrel 5524 Fun wfun 6318 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-fun 6326 |
This theorem is referenced by: (None) |
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