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Theorem iundomg 9360
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 9359 . . . 4 (𝜑𝑇 ≼ (𝐴 × 𝐶))
5 iundomg.4 . . . 4 (𝜑 → (𝐴 × 𝐶) ∈ AC 𝑥𝐴 𝐵)
6 acndom2 8874 . . . 4 (𝑇 ≼ (𝐴 × 𝐶) → ((𝐴 × 𝐶) ∈ AC 𝑥𝐴 𝐵𝑇AC 𝑥𝐴 𝐵))
74, 5, 6sylc 65 . . 3 (𝜑𝑇AC 𝑥𝐴 𝐵)
81iunfo 9358 . . 3 (2nd𝑇):𝑇onto 𝑥𝐴 𝐵
9 fodomacn 8876 . . 3 (𝑇AC 𝑥𝐴 𝐵 → ((2nd𝑇):𝑇onto 𝑥𝐴 𝐵 𝑥𝐴 𝐵𝑇))
107, 8, 9mpisyl 21 . 2 (𝜑 𝑥𝐴 𝐵𝑇)
11 domtr 8006 . 2 (( 𝑥𝐴 𝐵𝑇𝑇 ≼ (𝐴 × 𝐶)) → 𝑥𝐴 𝐵 ≼ (𝐴 × 𝐶))
1210, 4, 11syl2anc 693 1 (𝜑 𝑥𝐴 𝐵 ≼ (𝐴 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  wcel 1989  wral 2911  {csn 4175   ciun 4518   class class class wbr 4651   × cxp 5110  cres 5114  ontowfo 5884  (class class class)co 6647  2nd c2nd 7164  𝑚 cmap 7854  cdom 7950  AC wacn 8761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-1st 7165  df-2nd 7166  df-map 7856  df-dom 7954  df-acn 8765
This theorem is referenced by:  iundom  9361  iunctb  9393
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