Theorem uniimadomf 10458

Description: An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 10457 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 1921	. . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥) ≼ 𝐵
2		uniimadomf.1	. . . . 5 ⊢ Ⅎ𝑥𝐹
3		nfcv 2901	. . . . 5 ⊢ Ⅎ𝑥𝑧
4	2, 3	nffv 6837	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧)
5		nfcv 2901	. . . 4 ⊢ Ⅎ𝑥 ≼
6		nfcv 2901	. . . 4 ⊢ Ⅎ𝑥𝐵
7	4, 5, 6	nfbr 5119	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧) ≼ 𝐵
8		fveq2 6827	. . . 4 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
9	8	breq1d 5082	. . 3 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) ≼ 𝐵 ↔ (𝐹‘𝑧) ≼ 𝐵))
10	1, 7, 9	cbvralw 3281	. 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 10457	. 2 ⊢ ((Fun 𝐹 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ≼ 𝐵) → ∪ (𝐹 “ 𝐴) ≼ (𝐴 × 𝐵))
14	10, 13	sylan2b 600	1 ⊢ ((Fun 𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵) → ∪ (𝐹 “ 𝐴) ≼ (𝐴 × 𝐵))

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2119  Ⅎwnfc 2886  ∀wral 3053  Vcvv 3431  ∪ cuni 4838   class class class wbr 5072   × cxp 5616   “ cima 5621  Fun wfun 6479  ‘cfv 6485   ≼ cdom 8881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-ac2 10376

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-isom 6494  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-er 8633  df-map 8765  df-en 8884  df-dom 8885  df-card 9854  df-acn 9857  df-ac 10029

This theorem is referenced by: (None)