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Mirrors > Home > MPE Home > Th. List > imafiALT | Structured version Visualization version GIF version |
Description: Shorter proof of imafi 8842 using ax-pow 5247. (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 6086 | . 2 ⊢ (𝐹 “ dom (𝐹 ↾ 𝑋)) = (𝐹 “ 𝑋) | |
2 | simpr 488 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → 𝑋 ∈ Fin) | |
3 | dmres 5862 | . . . . 5 ⊢ dom (𝐹 ↾ 𝑋) = (𝑋 ∩ dom 𝐹) | |
4 | inss1 4133 | . . . . 5 ⊢ (𝑋 ∩ dom 𝐹) ⊆ 𝑋 | |
5 | 3, 4 | eqsstri 3925 | . . . 4 ⊢ dom (𝐹 ↾ 𝑋) ⊆ 𝑋 |
6 | ssfi 8840 | . . . 4 ⊢ ((𝑋 ∈ Fin ∧ dom (𝐹 ↾ 𝑋) ⊆ 𝑋) → dom (𝐹 ↾ 𝑋) ∈ Fin) | |
7 | 2, 5, 6 | sylancl 589 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → dom (𝐹 ↾ 𝑋) ∈ Fin) |
8 | resss 5865 | . . . . 5 ⊢ (𝐹 ↾ 𝑋) ⊆ 𝐹 | |
9 | dmss 5760 | . . . . 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 6632 | . . . 4 ⊢ ((Fun 𝐹 ∧ dom (𝐹 ↾ 𝑋) ⊆ dom 𝐹) → (𝐹 ↾ dom (𝐹 ↾ 𝑋)):dom (𝐹 ↾ 𝑋)–onto→(𝐹 “ dom (𝐹 ↾ 𝑋))) | |
12 | 10, 11 | syldan 594 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → (𝐹 ↾ dom (𝐹 ↾ 𝑋)):dom (𝐹 ↾ 𝑋)–onto→(𝐹 “ dom (𝐹 ↾ 𝑋))) |
13 | fofi 8951 | . . 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 587 | . 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → (𝐹 “ dom (𝐹 ↾ 𝑋)) ∈ Fin) |
15 | 1, 14 | eqeltrrid 2839 | 1 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → (𝐹 “ 𝑋) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2110 ∩ cin 3856 ⊆ wss 3857 dom cdm 5540 ↾ cres 5542 “ cima 5543 Fun wfun 6363 –onto→wfo 6367 Fincfn 8615 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-om 7634 df-1o 8191 df-er 8380 df-en 8616 df-dom 8617 df-fin 8619 |
This theorem is referenced by: (None) |
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