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 9956 | . . . 4 ⊢ (𝜑 → 𝑇 ≼ (𝐴 × 𝐶)) |
5 | iundomg.4 | . . . 4 ⊢ (𝜑 → (𝐴 × 𝐶) ∈ AC ∪ 𝑥 ∈ 𝐴 𝐵) | |
6 | acndom2 9474 | . . . 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 9955 | . . 3 ⊢ (2nd ↾ 𝑇):𝑇–onto→∪ 𝑥 ∈ 𝐴 𝐵 |
9 | fodomacn 9476 | . . 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 8556 | . 2 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ≼ 𝑇 ∧ 𝑇 ≼ (𝐴 × 𝐶)) → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ (𝐴 × 𝐶)) | |
12 | 10, 4, 11 | syl2anc 586 | 1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ (𝐴 × 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∀wral 3138 {csn 4560 ∪ ciun 4911 class class class wbr 5058 × cxp 5547 ↾ cres 5551 –onto→wfo 6347 (class class class)co 7150 2nd c2nd 7682 ↑m cmap 8400 ≼ cdom 8501 AC wacn 9361 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-map 8402 df-dom 8505 df-acn 9365 |
This theorem is referenced by: iundom 9958 iunctb 9990 |
Copyright terms: Public domain | W3C validator |