![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > iundomg | Structured version Visualization version GIF version |
Description: An upper bound for the cardinality of an indexed union, with explicit choice principles. 𝐵 depends on 𝑥 and should be thought of as 𝐵(𝑥). (Contributed by Mario Carneiro, 1-Sep-2015.) |
Ref | Expression |
---|---|
iunfo.1 | ⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) |
iundomg.2 | ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵) ∈ AC 𝐴) |
iundomg.3 | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ≼ 𝐶) |
iundomg.4 | ⊢ (𝜑 → (𝐴 × 𝐶) ∈ AC ∪ 𝑥 ∈ 𝐴 𝐵) |
Ref | Expression |
---|---|
iundomg | ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ (𝐴 × 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunfo.1 | . . . . 5 ⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) | |
2 | iundomg.2 | . . . . 5 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵) ∈ AC 𝐴) | |
3 | iundomg.3 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ≼ 𝐶) | |
4 | 1, 2, 3 | iundom2g 10541 | . . . 4 ⊢ (𝜑 → 𝑇 ≼ (𝐴 × 𝐶)) |
5 | iundomg.4 | . . . 4 ⊢ (𝜑 → (𝐴 × 𝐶) ∈ AC ∪ 𝑥 ∈ 𝐴 𝐵) | |
6 | acndom2 10055 | . . . 4 ⊢ (𝑇 ≼ (𝐴 × 𝐶) → ((𝐴 × 𝐶) ∈ AC ∪ 𝑥 ∈ 𝐴 𝐵 → 𝑇 ∈ AC ∪ 𝑥 ∈ 𝐴 𝐵)) | |
7 | 4, 5, 6 | sylc 65 | . . 3 ⊢ (𝜑 → 𝑇 ∈ AC ∪ 𝑥 ∈ 𝐴 𝐵) |
8 | 1 | iunfo 10540 | . . 3 ⊢ (2nd ↾ 𝑇):𝑇–onto→∪ 𝑥 ∈ 𝐴 𝐵 |
9 | fodomacn 10057 | . . 3 ⊢ (𝑇 ∈ AC ∪ 𝑥 ∈ 𝐴 𝐵 → ((2nd ↾ 𝑇):𝑇–onto→∪ 𝑥 ∈ 𝐴 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ 𝑇)) | |
10 | 7, 8, 9 | mpisyl 21 | . 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ 𝑇) |
11 | domtr 9009 | . 2 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ≼ 𝑇 ∧ 𝑇 ≼ (𝐴 × 𝐶)) → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ (𝐴 × 𝐶)) | |
12 | 10, 4, 11 | syl2anc 583 | 1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ (𝐴 × 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ∀wral 3060 {csn 4628 ∪ ciun 4997 class class class wbr 5148 × cxp 5674 ↾ cres 5678 –onto→wfo 6541 (class class class)co 7412 2nd c2nd 7978 ↑m cmap 8826 ≼ cdom 8943 AC wacn 9939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-map 8828 df-dom 8947 df-acn 9943 |
This theorem is referenced by: iundom 10543 iunctb 10575 |
Copyright terms: Public domain | W3C validator |