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Theorem iundomg 10533
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 10532 . . . 4 (𝜑𝑇 ≼ (𝐴 × 𝐶))
5 iundomg.4 . . . 4 (𝜑 → (𝐴 × 𝐶) ∈ AC 𝑥𝐴 𝐵)
6 acndom2 10046 . . . 4 (𝑇 ≼ (𝐴 × 𝐶) → ((𝐴 × 𝐶) ∈ AC 𝑥𝐴 𝐵𝑇AC 𝑥𝐴 𝐵))
74, 5, 6sylc 65 . . 3 (𝜑𝑇AC 𝑥𝐴 𝐵)
81iunfo 10531 . . 3 (2nd𝑇):𝑇onto 𝑥𝐴 𝐵
9 fodomacn 10048 . . 3 (𝑇AC 𝑥𝐴 𝐵 → ((2nd𝑇):𝑇onto 𝑥𝐴 𝐵 𝑥𝐴 𝐵𝑇))
107, 8, 9mpisyl 21 . 2 (𝜑 𝑥𝐴 𝐵𝑇)
11 domtr 9000 . 2 (( 𝑥𝐴 𝐵𝑇𝑇 ≼ (𝐴 × 𝐶)) → 𝑥𝐴 𝐵 ≼ (𝐴 × 𝐶))
1210, 4, 11syl2anc 585 1 (𝜑 𝑥𝐴 𝐵 ≼ (𝐴 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wral 3062  {csn 4628   ciun 4997   class class class wbr 5148   × cxp 5674  cres 5678  ontowfo 6539  (class class class)co 7406  2nd c2nd 7971  m cmap 8817  cdom 8934  AC wacn 9930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  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 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-1st 7972  df-2nd 7973  df-map 8819  df-dom 8938  df-acn 9934
This theorem is referenced by:  iundom  10534  iunctb  10566
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