Theorem iundomg 10581

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.)

Hypotheses

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)

Assertion

Ref	Expression
iundomg	⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ (𝐴 × 𝐶))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝑇(𝑥)

Proof of Theorem iundomg

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 10580	. . . 4 ⊢ (𝜑 → 𝑇 ≼ (𝐴 × 𝐶))
5		iundomg.4	. . . 4 ⊢ (𝜑 → (𝐴 × 𝐶) ∈ AC ∪ 𝑥 ∈ 𝐴 𝐵)
6		acndom2 10094	. . . 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 10579	. . 3 ⊢ (2nd ↾ 𝑇):𝑇–onto→∪ 𝑥 ∈ 𝐴 𝐵
9		fodomacn 10096	. . 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 9047	. 2 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ≼ 𝑇 ∧ 𝑇 ≼ (𝐴 × 𝐶)) → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ (𝐴 × 𝐶))
12	10, 4, 11	syl2anc 584	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ (𝐴 × 𝐶))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ∀wral 3061  {csn 4626  ∪ ciun 4991   class class class wbr 5143   × cxp 5683   ↾ cres 5687  –onto→wfo 6559  (class class class)co 7431  2^nd c2nd 8013   ↑_m cmap 8866   ≼ cdom 8983  AC wacn 9978

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-dom 8987  df-acn 9982

This theorem is referenced by:  iundom  10582  iunctb  10614