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Mirrors > Home > MPE Home > Th. List > imafiALT | Structured version Visualization version GIF version |
Description: Shorter proof of imafi 9177 using ax-pow 5356. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
imafiALT | ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → (𝐹 “ 𝑋) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imadmres 6227 | . 2 ⊢ (𝐹 “ dom (𝐹 ↾ 𝑋)) = (𝐹 “ 𝑋) | |
2 | simpr 484 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → 𝑋 ∈ Fin) | |
3 | dmres 5997 | . . . . 5 ⊢ dom (𝐹 ↾ 𝑋) = (𝑋 ∩ dom 𝐹) | |
4 | inss1 4223 | . . . . 5 ⊢ (𝑋 ∩ dom 𝐹) ⊆ 𝑋 | |
5 | 3, 4 | eqsstri 4011 | . . . 4 ⊢ dom (𝐹 ↾ 𝑋) ⊆ 𝑋 |
6 | ssfi 9175 | . . . 4 ⊢ ((𝑋 ∈ Fin ∧ dom (𝐹 ↾ 𝑋) ⊆ 𝑋) → dom (𝐹 ↾ 𝑋) ∈ Fin) | |
7 | 2, 5, 6 | sylancl 585 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → dom (𝐹 ↾ 𝑋) ∈ Fin) |
8 | resss 6000 | . . . . 5 ⊢ (𝐹 ↾ 𝑋) ⊆ 𝐹 | |
9 | dmss 5896 | . . . . 5 ⊢ ((𝐹 ↾ 𝑋) ⊆ 𝐹 → dom (𝐹 ↾ 𝑋) ⊆ dom 𝐹) | |
10 | 8, 9 | mp1i 13 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → dom (𝐹 ↾ 𝑋) ⊆ dom 𝐹) |
11 | fores 6809 | . . . 4 ⊢ ((Fun 𝐹 ∧ dom (𝐹 ↾ 𝑋) ⊆ dom 𝐹) → (𝐹 ↾ dom (𝐹 ↾ 𝑋)):dom (𝐹 ↾ 𝑋)–onto→(𝐹 “ dom (𝐹 ↾ 𝑋))) | |
12 | 10, 11 | syldan 590 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → (𝐹 ↾ dom (𝐹 ↾ 𝑋)):dom (𝐹 ↾ 𝑋)–onto→(𝐹 “ dom (𝐹 ↾ 𝑋))) |
13 | fofi 9340 | . . 3 ⊢ ((dom (𝐹 ↾ 𝑋) ∈ Fin ∧ (𝐹 ↾ dom (𝐹 ↾ 𝑋)):dom (𝐹 ↾ 𝑋)–onto→(𝐹 “ dom (𝐹 ↾ 𝑋))) → (𝐹 “ dom (𝐹 ↾ 𝑋)) ∈ Fin) | |
14 | 7, 12, 13 | syl2anc 583 | . 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → (𝐹 “ dom (𝐹 ↾ 𝑋)) ∈ Fin) |
15 | 1, 14 | eqeltrrid 2832 | 1 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → (𝐹 “ 𝑋) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2098 ∩ cin 3942 ⊆ wss 3943 dom cdm 5669 ↾ cres 5671 “ cima 5672 Fun wfun 6531 –onto→wfo 6535 Fincfn 8941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-om 7853 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-fin 8945 |
This theorem is referenced by: (None) |
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