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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 = 𝑥𝐴 ({𝑥} × 𝐵)
iundomg.2 (𝜑 𝑥𝐴 (𝐶m 𝐵) ∈ AC 𝐴)
iundomg.3 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
iundomg.4 (𝜑 → (𝐴 × 𝐶) ∈ AC 𝑥𝐴 𝐵)
Assertion
Ref Expression
iundomg (𝜑 𝑥𝐴 𝐵 ≼ (𝐴 × 𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝑇(𝑥)

Proof of Theorem iundomg
StepHypRef Expression
1 iunfo.1 . . . . 5 𝑇 = 𝑥𝐴 ({𝑥} × 𝐵)
2 iundomg.2 . . . . 5 (𝜑 𝑥𝐴 (𝐶m 𝐵) ∈ AC 𝐴)
3 iundomg.3 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
41, 2, 3iundom2g 10580 . . . 4 (𝜑𝑇 ≼ (𝐴 × 𝐶))
5 iundomg.4 . . . 4 (𝜑 → (𝐴 × 𝐶) ∈ AC 𝑥𝐴 𝐵)
6 acndom2 10094 . . . 4 (𝑇 ≼ (𝐴 × 𝐶) → ((𝐴 × 𝐶) ∈ AC 𝑥𝐴 𝐵𝑇AC 𝑥𝐴 𝐵))
74, 5, 6sylc 65 . . 3 (𝜑𝑇AC 𝑥𝐴 𝐵)
81iunfo 10579 . . 3 (2nd𝑇):𝑇onto 𝑥𝐴 𝐵
9 fodomacn 10096 . . 3 (𝑇AC 𝑥𝐴 𝐵 → ((2nd𝑇):𝑇onto 𝑥𝐴 𝐵 𝑥𝐴 𝐵𝑇))
107, 8, 9mpisyl 21 . 2 (𝜑 𝑥𝐴 𝐵𝑇)
11 domtr 9047 . 2 (( 𝑥𝐴 𝐵𝑇𝑇 ≼ (𝐴 × 𝐶)) → 𝑥𝐴 𝐵 ≼ (𝐴 × 𝐶))
1210, 4, 11syl2anc 584 1 (𝜑 𝑥𝐴 𝐵 ≼ (𝐴 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wral 3061  {csn 4626   ciun 4991   class class class wbr 5143   × cxp 5683  cres 5687  ontowfo 6559  (class class class)co 7431  2nd 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
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