Theorem uniimadomf 10142

Description: An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 10141 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 1922	. . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥) ≼ 𝐵
2		uniimadomf.1	. . . . 5 ⊢ Ⅎ𝑥𝐹
3		nfcv 2900	. . . . 5 ⊢ Ⅎ𝑥𝑧
4	2, 3	nffv 6716	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧)
5		nfcv 2900	. . . 4 ⊢ Ⅎ𝑥 ≼
6		nfcv 2900	. . . 4 ⊢ Ⅎ𝑥𝐵
7	4, 5, 6	nfbr 5090	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧) ≼ 𝐵
8		fveq2 6706	. . . 4 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
9	8	breq1d 5053	. . 3 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) ≼ 𝐵 ↔ (𝐹‘𝑧) ≼ 𝐵))
10	1, 7, 9	cbvralw 3342	. 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 10141	. 2 ⊢ ((Fun 𝐹 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ≼ 𝐵) → ∪ (𝐹 “ 𝐴) ≼ (𝐴 × 𝐵))
14	10, 13	sylan2b 597	1 ⊢ ((Fun 𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵) → ∪ (𝐹 “ 𝐴) ≼ (𝐴 × 𝐵))

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2110  Ⅎwnfc 2880  ∀wral 3054  Vcvv 3401  ∪ cuni 4809   class class class wbr 5043   × cxp 5538   “ cima 5543  Fun wfun 6363  ‘cfv 6369   ≼ cdom 8613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-rep 5168  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512  ax-ac2 10060

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-int 4850  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-se 5499  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-ord 6205  df-on 6206  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-isom 6378  df-riota 7159  df-ov 7205  df-oprab 7206  df-mpo 7207  df-1st 7750  df-2nd 7751  df-wrecs 8036  df-recs 8097  df-er 8380  df-map 8499  df-en 8616  df-dom 8617  df-card 9538  df-acn 9541  df-ac 9713

This theorem is referenced by: (None)