Theorem tz7.48-2 8381

Description: Proposition 7.48(2) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.) (Revised by David Abernethy, 5-May-2013.)

Hypothesis

Ref	Expression
tz7.48.1	⊢ 𝐹 Fn On

Assertion

Ref	Expression
tz7.48-2	⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → Fun ◡𝐹)

Distinct variable group:   𝑥,𝐹

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem tz7.48-2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3944	. . 3 ⊢ On ⊆ On
2		onelon 6348	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
3	2	ancoms 458	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ On) → 𝑦 ∈ On)
4		tz7.48.1	. . . . . . . . . . 11 ⊢ 𝐹 Fn On
5	4	fndmi 6602	. . . . . . . . . 10 ⊢ dom 𝐹 = On
6	5	eleq2i 2828	. . . . . . . . 9 ⊢ (𝑦 ∈ dom 𝐹 ↔ 𝑦 ∈ On)
7		fnfun 6598	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn On → Fun 𝐹)
8	4, 7	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 𝐹
9		funfvima 7185	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹 “ 𝑥)))
10	8, 9	mpan 691	. . . . . . . . . . 11 ⊢ (𝑦 ∈ dom 𝐹 → (𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹 “ 𝑥)))
11	10	impcom 407	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) ∈ (𝐹 “ 𝑥))
12		eleq1a 2831	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑦) ∈ (𝐹 “ 𝑥) → ((𝐹‘𝑥) = (𝐹‘𝑦) → (𝐹‘𝑥) ∈ (𝐹 “ 𝑥)))
13		eldifn 4072	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ (𝐹‘𝑥) ∈ (𝐹 “ 𝑥))
14	12, 13	nsyli 157	. . . . . . . . . 10 ⊢ ((𝐹‘𝑦) ∈ (𝐹 “ 𝑥) → ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
15	11, 14	syl 17	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 ∈ dom 𝐹) → ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
16	6, 15	sylan2br 596	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 ∈ On) → ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
17	3, 16	syldan 592	. . . . . . 7 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ On) → ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
18	17	expimpd 453	. . . . . 6 ⊢ (𝑦 ∈ 𝑥 → ((𝑥 ∈ On ∧ (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
19	18	com12 32	. . . . 5 ⊢ ((𝑥 ∈ On ∧ (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) → (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
20	19	ralrimiv 3128	. . . 4 ⊢ ((𝑥 ∈ On ∧ (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) → ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦))
21	20	ralimiaa 3073	. . 3 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦))
22	4	tz7.48lem 8380	. . 3 ⊢ ((On ⊆ On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → Fun ◡(𝐹 ↾ On))
23	1, 21, 22	sylancr 588	. 2 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → Fun ◡(𝐹 ↾ On))
24		fnrel 6600	. . . . . 6 ⊢ (𝐹 Fn On → Rel 𝐹)
25	4, 24	ax-mp 5	. . . . 5 ⊢ Rel 𝐹
26	5	eqimssi 3982	. . . . 5 ⊢ dom 𝐹 ⊆ On
27		relssres 5987	. . . . 5 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
28	25, 26, 27	mp2an 693	. . . 4 ⊢ (𝐹 ↾ On) = 𝐹
29	28	cnveqi 5829	. . 3 ⊢ ◡(𝐹 ↾ On) = ◡𝐹
30	29	funeqi 6519	. 2 ⊢ (Fun ◡(𝐹 ↾ On) ↔ Fun ◡𝐹)
31	23, 30	sylib 218	1 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → Fun ◡𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051   ∖ cdif 3886   ⊆ wss 3889  ^◡ccnv 5630  dom cdm 5631   ↾ cres 5633   “ cima 5634  Rel wrel 5636  Oncon0 6323  Fun wfun 6492   Fn wfn 6493  ‘cfv 6498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fv 6506

This theorem is referenced by:  tz7.48-3  8383  onvf1od  35289