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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rn1st | Structured version Visualization version GIF version |
Description: The range of a function with a first-countable domain is itself first-countable. This is a variation of 1stcrestlem 23476, with a not-free hypothesis replacing a disjoint variable constraint. (Contributed by Glauco Siliprandi, 24-Jan-2025.) |
Ref | Expression |
---|---|
rn1st.1 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
rn1st | ⊢ (𝐵 ≼ ω → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordom 7897 | . . . . . 6 ⊢ Ord ω | |
2 | reldom 8990 | . . . . . . . 8 ⊢ Rel ≼ | |
3 | 2 | brrelex2i 5746 | . . . . . . 7 ⊢ (𝐵 ≼ ω → ω ∈ V) |
4 | elong 6394 | . . . . . . 7 ⊢ (ω ∈ V → (ω ∈ On ↔ Ord ω)) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝐵 ≼ ω → (ω ∈ On ↔ Ord ω)) |
6 | 1, 5 | mpbiri 258 | . . . . 5 ⊢ (𝐵 ≼ ω → ω ∈ On) |
7 | ondomen 10075 | . . . . 5 ⊢ ((ω ∈ On ∧ 𝐵 ≼ ω) → 𝐵 ∈ dom card) | |
8 | 6, 7 | mpancom 688 | . . . 4 ⊢ (𝐵 ≼ ω → 𝐵 ∈ dom card) |
9 | rn1st.1 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
10 | eqid 2735 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶) | |
11 | 9, 10 | dmmptssf 45175 | . . . 4 ⊢ dom (𝑥 ∈ 𝐵 ↦ 𝐶) ⊆ 𝐵 |
12 | ssnum 10077 | . . . 4 ⊢ ((𝐵 ∈ dom card ∧ dom (𝑥 ∈ 𝐵 ↦ 𝐶) ⊆ 𝐵) → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ dom card) | |
13 | 8, 11, 12 | sylancl 586 | . . 3 ⊢ (𝐵 ≼ ω → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ dom card) |
14 | funmpt 6606 | . . . 4 ⊢ Fun (𝑥 ∈ 𝐵 ↦ 𝐶) | |
15 | funforn 6828 | . . . 4 ⊢ (Fun (𝑥 ∈ 𝐵 ↦ 𝐶) ↔ (𝑥 ∈ 𝐵 ↦ 𝐶):dom (𝑥 ∈ 𝐵 ↦ 𝐶)–onto→ran (𝑥 ∈ 𝐵 ↦ 𝐶)) | |
16 | 14, 15 | mpbi 230 | . . 3 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶):dom (𝑥 ∈ 𝐵 ↦ 𝐶)–onto→ran (𝑥 ∈ 𝐵 ↦ 𝐶) |
17 | fodomnum 10095 | . . 3 ⊢ (dom (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ dom card → ((𝑥 ∈ 𝐵 ↦ 𝐶):dom (𝑥 ∈ 𝐵 ↦ 𝐶)–onto→ran (𝑥 ∈ 𝐵 ↦ 𝐶) → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ dom (𝑥 ∈ 𝐵 ↦ 𝐶))) | |
18 | 13, 16, 17 | mpisyl 21 | . 2 ⊢ (𝐵 ≼ ω → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ dom (𝑥 ∈ 𝐵 ↦ 𝐶)) |
19 | ctex 9003 | . . . 4 ⊢ (𝐵 ≼ ω → 𝐵 ∈ V) | |
20 | ssdomg 9039 | . . . 4 ⊢ (𝐵 ∈ V → (dom (𝑥 ∈ 𝐵 ↦ 𝐶) ⊆ 𝐵 → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ 𝐵)) | |
21 | 19, 11, 20 | mpisyl 21 | . . 3 ⊢ (𝐵 ≼ ω → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ 𝐵) |
22 | domtr 9046 | . . 3 ⊢ ((dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ 𝐵 ∧ 𝐵 ≼ ω) → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω) | |
23 | 21, 22 | mpancom 688 | . 2 ⊢ (𝐵 ≼ ω → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω) |
24 | domtr 9046 | . 2 ⊢ ((ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ dom (𝑥 ∈ 𝐵 ↦ 𝐶) ∧ dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω) → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω) | |
25 | 18, 23, 24 | syl2anc 584 | 1 ⊢ (𝐵 ≼ ω → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2106 Ⅎwnfc 2888 Vcvv 3478 ⊆ wss 3963 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5689 ran crn 5690 Ord word 6385 Oncon0 6386 Fun wfun 6557 –onto→wfo 6561 ωcom 7887 ≼ cdom 8982 cardccrd 9973 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-card 9977 df-acn 9980 |
This theorem is referenced by: saliunclf 46278 |
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