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Theorem uniimadomf 9569
Description: An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 9568 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 1995 . . 3 𝑧(𝐹𝑥) ≼ 𝐵
2 uniimadomf.1 . . . . 5 𝑥𝐹
3 nfcv 2913 . . . . 5 𝑥𝑧
42, 3nffv 6339 . . . 4 𝑥(𝐹𝑧)
5 nfcv 2913 . . . 4 𝑥
6 nfcv 2913 . . . 4 𝑥𝐵
74, 5, 6nfbr 4833 . . 3 𝑥(𝐹𝑧) ≼ 𝐵
8 fveq2 6332 . . . 4 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
98breq1d 4796 . . 3 (𝑥 = 𝑧 → ((𝐹𝑥) ≼ 𝐵 ↔ (𝐹𝑧) ≼ 𝐵))
101, 7, 9cbvral 3316 . 2 (∀𝑥𝐴 (𝐹𝑥) ≼ 𝐵 ↔ ∀𝑧𝐴 (𝐹𝑧) ≼ 𝐵)
11 uniimadomf.2 . . 3 𝐴 ∈ V
12 uniimadomf.3 . . 3 𝐵 ∈ V
1311, 12uniimadom 9568 . 2 ((Fun 𝐹 ∧ ∀𝑧𝐴 (𝐹𝑧) ≼ 𝐵) → (𝐹𝐴) ≼ (𝐴 × 𝐵))
1410, 13sylan2b 581 1 ((Fun 𝐹 ∧ ∀𝑥𝐴 (𝐹𝑥) ≼ 𝐵) → (𝐹𝐴) ≼ (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 2145  wnfc 2900  wral 3061  Vcvv 3351   cuni 4574   class class class wbr 4786   × cxp 5247  cima 5252  Fun wfun 6025  cfv 6031  cdom 8107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-ac2 9487
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-card 8965  df-acn 8968  df-ac 9139
This theorem is referenced by: (None)
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