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Theorem uniimadomf 9763
Description: An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 9762 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by NM, 26-Mar-2006.)
Hypotheses
Ref Expression
uniimadomf.1 𝑥𝐹
uniimadomf.2 𝐴 ∈ V
uniimadomf.3 𝐵 ∈ V
Assertion
Ref Expression
uniimadomf ((Fun 𝐹 ∧ ∀𝑥𝐴 (𝐹𝑥) ≼ 𝐵) → (𝐹𝐴) ≼ (𝐴 × 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem uniimadomf
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1874 . . 3 𝑧(𝐹𝑥) ≼ 𝐵
2 uniimadomf.1 . . . . 5 𝑥𝐹
3 nfcv 2925 . . . . 5 𝑥𝑧
42, 3nffv 6506 . . . 4 𝑥(𝐹𝑧)
5 nfcv 2925 . . . 4 𝑥
6 nfcv 2925 . . . 4 𝑥𝐵
74, 5, 6nfbr 4972 . . 3 𝑥(𝐹𝑧) ≼ 𝐵
8 fveq2 6496 . . . 4 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
98breq1d 4935 . . 3 (𝑥 = 𝑧 → ((𝐹𝑥) ≼ 𝐵 ↔ (𝐹𝑧) ≼ 𝐵))
101, 7, 9cbvral 3372 . 2 (∀𝑥𝐴 (𝐹𝑥) ≼ 𝐵 ↔ ∀𝑧𝐴 (𝐹𝑧) ≼ 𝐵)
11 uniimadomf.2 . . 3 𝐴 ∈ V
12 uniimadomf.3 . . 3 𝐵 ∈ V
1311, 12uniimadom 9762 . 2 ((Fun 𝐹 ∧ ∀𝑧𝐴 (𝐹𝑧) ≼ 𝐵) → (𝐹𝐴) ≼ (𝐴 × 𝐵))
1410, 13sylan2b 585 1 ((Fun 𝐹 ∧ ∀𝑥𝐴 (𝐹𝑥) ≼ 𝐵) → (𝐹𝐴) ≼ (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  wcel 2051  wnfc 2909  wral 3081  Vcvv 3408   cuni 4708   class class class wbr 4925   × cxp 5401  cima 5406  Fun wfun 6179  cfv 6185  cdom 8302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277  ax-ac2 9681
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rmo 3089  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-se 5363  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-isom 6194  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-1st 7499  df-2nd 7500  df-wrecs 7748  df-recs 7810  df-er 8087  df-map 8206  df-en 8305  df-dom 8306  df-card 9160  df-acn 9163  df-ac 9334
This theorem is referenced by: (None)
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