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Theorem iundomg 9242
 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 9241 . . . 4 (𝜑𝑇 ≼ (𝐴 × 𝐶))
5 iundomg.4 . . . 4 (𝜑 → (𝐴 × 𝐶) ∈ AC 𝑥𝐴 𝐵)
6 acndom2 8760 . . . 4 (𝑇 ≼ (𝐴 × 𝐶) → ((𝐴 × 𝐶) ∈ AC 𝑥𝐴 𝐵𝑇AC 𝑥𝐴 𝐵))
74, 5, 6sylc 63 . . 3 (𝜑𝑇AC 𝑥𝐴 𝐵)
81iunfo 9240 . . 3 (2nd𝑇):𝑇onto 𝑥𝐴 𝐵
9 fodomacn 8762 . . 3 (𝑇AC 𝑥𝐴 𝐵 → ((2nd𝑇):𝑇onto 𝑥𝐴 𝐵 𝑥𝐴 𝐵𝑇))
107, 8, 9mpisyl 21 . 2 (𝜑 𝑥𝐴 𝐵𝑇)
11 domtr 7895 . 2 (( 𝑥𝐴 𝐵𝑇𝑇 ≼ (𝐴 × 𝐶)) → 𝑥𝐴 𝐵 ≼ (𝐴 × 𝐶))
1210, 4, 11syl2anc 691 1 (𝜑 𝑥𝐴 𝐵 ≼ (𝐴 × 𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {csn 4125  ∪ ciun 4455   class class class wbr 4583   × cxp 5036   ↾ cres 5040  –onto→wfo 5802  (class class class)co 6549  2nd c2nd 7058   ↑𝑚 cmap 7744   ≼ cdom 7839  AC wacn 8647 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-dom 7843  df-acn 8651 This theorem is referenced by:  iundom  9243  iunctb  9275
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