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Theorem uniimadomf 9223
Description: An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 9222 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 1829 . . 3 𝑧(𝐹𝑥) ≼ 𝐵
2 uniimadomf.1 . . . . 5 𝑥𝐹
3 nfcv 2750 . . . . 5 𝑥𝑧
42, 3nffv 6095 . . . 4 𝑥(𝐹𝑧)
5 nfcv 2750 . . . 4 𝑥
6 nfcv 2750 . . . 4 𝑥𝐵
74, 5, 6nfbr 4623 . . 3 𝑥(𝐹𝑧) ≼ 𝐵
8 fveq2 6088 . . . 4 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
98breq1d 4587 . . 3 (𝑥 = 𝑧 → ((𝐹𝑥) ≼ 𝐵 ↔ (𝐹𝑧) ≼ 𝐵))
101, 7, 9cbvral 3142 . 2 (∀𝑥𝐴 (𝐹𝑥) ≼ 𝐵 ↔ ∀𝑧𝐴 (𝐹𝑧) ≼ 𝐵)
11 uniimadomf.2 . . 3 𝐴 ∈ V
12 uniimadomf.3 . . 3 𝐵 ∈ V
1311, 12uniimadom 9222 . 2 ((Fun 𝐹 ∧ ∀𝑧𝐴 (𝐹𝑧) ≼ 𝐵) → (𝐹𝐴) ≼ (𝐴 × 𝐵))
1410, 13sylan2b 490 1 ((Fun 𝐹 ∧ ∀𝑥𝐴 (𝐹𝑥) ≼ 𝐵) → (𝐹𝐴) ≼ (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 1976  wnfc 2737  wral 2895  Vcvv 3172   cuni 4366   class class class wbr 4577   × cxp 5026  cima 5031  Fun wfun 5784  cfv 5790  cdom 7816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-ac2 9145
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-er 7606  df-map 7723  df-en 7819  df-dom 7820  df-card 8625  df-acn 8628  df-ac 8799
This theorem is referenced by: (None)
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