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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 6543 | . 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | ctex 8909 | . . . 4 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) | |
3 | eqid 2733 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | dmmpt 6196 | . . . . 5 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
5 | df-rab 3407 | . . . . . 6 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V)} | |
6 | simpl 484 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V) → 𝑥 ∈ 𝐴) | |
7 | 6 | ss2abi 4027 | . . . . . . 7 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V)} ⊆ {𝑥 ∣ 𝑥 ∈ 𝐴} |
8 | mptctf.1 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
9 | 8 | abid2f 2936 | . . . . . . 7 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 |
10 | 7, 9 | sseqtri 3984 | . . . . . 6 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V)} ⊆ 𝐴 |
11 | 5, 10 | eqsstri 3982 | . . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ 𝐴 |
12 | 4, 11 | eqsstri 3982 | . . . 4 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴 |
13 | ssdomg 8946 | . . . 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 8953 | . . 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 6535 | . . 3 ⊢ (Fun (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn dom (𝑥 ∈ 𝐴 ↦ 𝐵)) | |
18 | fnct 10481 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵) Fn dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ≼ ω) → (𝑥 ∈ 𝐴 ↦ 𝐵) ≼ ω) | |
19 | 17, 18 | sylanb 582 | . 2 ⊢ ((Fun (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ≼ ω) → (𝑥 ∈ 𝐴 ↦ 𝐵) ≼ ω) |
20 | 1, 16, 19 | sylancr 588 | 1 ⊢ (𝐴 ≼ ω → (𝑥 ∈ 𝐴 ↦ 𝐵) ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 {cab 2710 Ⅎwnfc 2884 {crab 3406 Vcvv 3447 ⊆ wss 3914 class class class wbr 5109 ↦ cmpt 5192 dom cdm 5637 Fun wfun 6494 Fn wfn 6495 ωcom 7806 ≼ cdom 8887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-inf2 9585 ax-ac2 10407 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-oi 9454 df-card 9883 df-acn 9886 df-ac 10060 |
This theorem is referenced by: abrexctf 31689 |
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