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Mirrors > Home > MPE Home > Th. List > carduniima | Structured version Visualization version GIF version |
Description: The union of the image of a mapping to cardinals is a cardinal. Proposition 11.16 of [TakeutiZaring] p. 104. (Contributed by NM, 4-Nov-2004.) |
Ref | Expression |
---|---|
carduniima | ⊢ (𝐴 ∈ 𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → ∪ (𝐹 “ 𝐴) ∈ (ω ∪ ran ℵ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffun 5961 | . . . . 5 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → Fun 𝐹) | |
2 | funimaexg 5889 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝐵) → (𝐹 “ 𝐴) ∈ V) | |
3 | 1, 2 | sylan 487 | . . . 4 ⊢ ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ 𝐴 ∈ 𝐵) → (𝐹 “ 𝐴) ∈ V) |
4 | 3 | expcom 450 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹 “ 𝐴) ∈ V)) |
5 | ffn 5958 | . . . . . . . . 9 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → 𝐹 Fn 𝐴) | |
6 | fnima 5923 | . . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹) | |
7 | 5, 6 | syl 17 | . . . . . . . 8 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹 “ 𝐴) = ran 𝐹) |
8 | frn 5966 | . . . . . . . 8 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → ran 𝐹 ⊆ (ω ∪ ran ℵ)) | |
9 | 7, 8 | eqsstrd 3602 | . . . . . . 7 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹 “ 𝐴) ⊆ (ω ∪ ran ℵ)) |
10 | 9 | sseld 3567 | . . . . . 6 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥 ∈ (𝐹 “ 𝐴) → 𝑥 ∈ (ω ∪ ran ℵ))) |
11 | iscard3 8799 | . . . . . 6 ⊢ ((card‘𝑥) = 𝑥 ↔ 𝑥 ∈ (ω ∪ ran ℵ)) | |
12 | 10, 11 | syl6ibr 241 | . . . . 5 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝑥 ∈ (𝐹 “ 𝐴) → (card‘𝑥) = 𝑥)) |
13 | 12 | ralrimiv 2948 | . . . 4 ⊢ (𝐹:𝐴⟶(ω ∪ ran ℵ) → ∀𝑥 ∈ (𝐹 “ 𝐴)(card‘𝑥) = 𝑥) |
14 | carduni 8690 | . . . 4 ⊢ ((𝐹 “ 𝐴) ∈ V → (∀𝑥 ∈ (𝐹 “ 𝐴)(card‘𝑥) = 𝑥 → (card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴))) | |
15 | 13, 14 | syl5 33 | . . 3 ⊢ ((𝐹 “ 𝐴) ∈ V → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴))) |
16 | 4, 15 | syli 38 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴))) |
17 | iscard3 8799 | . 2 ⊢ ((card‘∪ (𝐹 “ 𝐴)) = ∪ (𝐹 “ 𝐴) ↔ ∪ (𝐹 “ 𝐴) ∈ (ω ∪ ran ℵ)) | |
18 | 16, 17 | syl6ib 240 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → ∪ (𝐹 “ 𝐴) ∈ (ω ∪ ran ℵ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ∪ cun 3538 ∪ cuni 4372 ran crn 5039 “ cima 5041 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 ωcom 6957 cardccrd 8644 ℵcale 8645 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-oi 8298 df-har 8346 df-card 8648 df-aleph 8649 |
This theorem is referenced by: cardinfima 8803 |
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