Theorem iundomg 9555

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 9554	. . . 4 ⊢ (𝜑 → 𝑇 ≼ (𝐴 × 𝐶))
5		iundomg.4	. . . 4 ⊢ (𝜑 → (𝐴 × 𝐶) ∈ AC ∪ 𝑥 ∈ 𝐴 𝐵)
6		acndom2 9067	. . . 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 9553	. . 3 ⊢ (2nd ↾ 𝑇):𝑇–onto→∪ 𝑥 ∈ 𝐴 𝐵
9		fodomacn 9069	. . 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 8174	. 2 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ≼ 𝑇 ∧ 𝑇 ≼ (𝐴 × 𝐶)) → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ (𝐴 × 𝐶))
12	10, 4, 11	syl2anc 696	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ (𝐴 × 𝐶))

Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139  ∀wral 3050  {csn 4321  ∪ ciun 4672   class class class wbr 4804   × cxp 5264   ↾ cres 5268  –onto→wfo 6047  (class class class)co 6813  2^nd c2nd 7332   ↑_𝑚 cmap 8023   ≼ cdom 8119  AC wacn 8954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025  df-dom 8123  df-acn 8958

This theorem is referenced by:  iundom  9556  iunctb  9588