Theorem uniimadomf 10439

Description: An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 10438 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by NM, 26-Mar-2006.)

Hypotheses

Ref	Expression
uniimadomf.1	⊢ Ⅎ𝑥𝐹
uniimadomf.2	⊢ 𝐴 ∈ V
uniimadomf.3	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
uniimadomf	⊢ ((Fun 𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵) → ∪ (𝐹 “ 𝐴) ≼ (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.

Step	Hyp	Ref	Expression
1		nfv 1914	. . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥) ≼ 𝐵
2		uniimadomf.1	. . . . 5 ⊢ Ⅎ𝑥𝐹
3		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑥𝑧
4	2, 3	nffv 6832	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧)
5		nfcv 2891	. . . 4 ⊢ Ⅎ𝑥 ≼
6		nfcv 2891	. . . 4 ⊢ Ⅎ𝑥𝐵
7	4, 5, 6	nfbr 5139	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧) ≼ 𝐵
8		fveq2 6822	. . . 4 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
9	8	breq1d 5102	. . 3 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) ≼ 𝐵 ↔ (𝐹‘𝑧) ≼ 𝐵))
10	1, 7, 9	cbvralw 3271	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵 ↔ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ≼ 𝐵)
11		uniimadomf.2	. . 3 ⊢ 𝐴 ∈ V
12		uniimadomf.3	. . 3 ⊢ 𝐵 ∈ V
13	11, 12	uniimadom 10438	. 2 ⊢ ((Fun 𝐹 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ≼ 𝐵) → ∪ (𝐹 “ 𝐴) ≼ (𝐴 × 𝐵))
14	10, 13	sylan2b 594	1 ⊢ ((Fun 𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵) → ∪ (𝐹 “ 𝐴) ≼ (𝐴 × 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  Ⅎwnfc 2876  ∀wral 3044  Vcvv 3436  ∪ cuni 4858   class class class wbr 5092   × cxp 5617   “ cima 5622  Fun wfun 6476  ‘cfv 6482   ≼ cdom 8870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-ac2 10357

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-card 9835  df-acn 9838  df-ac 10010

This theorem is referenced by: (None)