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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mptctf | Structured version Visualization version GIF version |
Description: A countable mapping set is countable, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Thierry Arnoux, 8-Mar-2017.) |
Ref | Expression |
---|---|
mptctf.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
mptctf | ⊢ (𝐴 ≼ ω → (𝑥 ∈ 𝐴 ↦ 𝐵) ≼ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funmpt 6618 | . 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | ctex 9025 | . . . 4 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) | |
3 | eqid 2740 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | dmmpt 6273 | . . . . 5 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
5 | df-rab 3444 | . . . . . 6 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V)} | |
6 | simpl 482 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V) → 𝑥 ∈ 𝐴) | |
7 | 6 | ss2abi 4090 | . . . . . . 7 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V)} ⊆ {𝑥 ∣ 𝑥 ∈ 𝐴} |
8 | mptctf.1 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
9 | 8 | abid2f 2942 | . . . . . . 7 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 |
10 | 7, 9 | sseqtri 4045 | . . . . . 6 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V)} ⊆ 𝐴 |
11 | 5, 10 | eqsstri 4043 | . . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ 𝐴 |
12 | 4, 11 | eqsstri 4043 | . . . 4 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴 |
13 | ssdomg 9062 | . . . 4 ⊢ (𝐴 ∈ V → (dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ≼ 𝐴)) | |
14 | 2, 12, 13 | mpisyl 21 | . . 3 ⊢ (𝐴 ≼ ω → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ≼ 𝐴) |
15 | domtr 9069 | . . 3 ⊢ ((dom (𝑥 ∈ 𝐴 ↦ 𝐵) ≼ 𝐴 ∧ 𝐴 ≼ ω) → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ≼ ω) | |
16 | 14, 15 | mpancom 687 | . 2 ⊢ (𝐴 ≼ ω → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ≼ ω) |
17 | funfn 6610 | . . 3 ⊢ (Fun (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn dom (𝑥 ∈ 𝐴 ↦ 𝐵)) | |
18 | fnct 10608 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵) Fn dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ≼ ω) → (𝑥 ∈ 𝐴 ↦ 𝐵) ≼ ω) | |
19 | 17, 18 | sylanb 580 | . 2 ⊢ ((Fun (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ≼ ω) → (𝑥 ∈ 𝐴 ↦ 𝐵) ≼ ω) |
20 | 1, 16, 19 | sylancr 586 | 1 ⊢ (𝐴 ≼ ω → (𝑥 ∈ 𝐴 ↦ 𝐵) ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 {cab 2717 Ⅎwnfc 2893 {crab 3443 Vcvv 3488 ⊆ wss 3976 class class class wbr 5166 ↦ cmpt 5249 dom cdm 5700 Fun wfun 6569 Fn wfn 6570 ωcom 7905 ≼ cdom 9003 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-inf2 9712 ax-ac2 10534 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-isom 6584 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-1st 8032 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-er 8765 df-map 8888 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-oi 9581 df-card 10010 df-acn 10013 df-ac 10187 |
This theorem is referenced by: abrexctf 32734 |
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