Theorem imasetpreimafvbijlemf1 46372

Description: Lemma for imasetpreimafvbij 46374: the mapping 𝐻 is an injective function into the range of function 𝐹. (Contributed by AV, 9-Mar-2024.) (Revised by AV, 22-Mar-2024.)

Hypotheses

Ref	Expression
fundcmpsurinj.p	⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
fundcmpsurinj.h	⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))

Assertion

Ref	Expression
imasetpreimafvbijlemf1	⊢ (𝐹 Fn 𝐴 → 𝐻:𝑃–1-1→(𝐹 “ 𝐴))

Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧,𝑝   𝑃,𝑝   𝐴,𝑝,𝑥,𝑧   𝑥,𝑃

Allowed substitution hints:   𝑃(𝑧)   𝐻(𝑥,𝑧,𝑝)

Proof of Theorem imasetpreimafvbijlemf1

Dummy variables 𝑎 𝑏 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fundcmpsurinj.p	. . 3 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
2		fundcmpsurinj.h	. . 3 ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))
3	1, 2	imasetpreimafvbijlemf 46369	. 2 ⊢ (𝐹 Fn 𝐴 → 𝐻:𝑃⟶(𝐹 “ 𝐴))
4	1, 2	imasetpreimafvbijlemfv1 46371	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑠 ∈ 𝑃) → ∃𝑏 ∈ 𝑠 (𝐻‘𝑠) = (𝐹‘𝑏))
5	1, 2	imasetpreimafvbijlemfv1 46371	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑟 ∈ 𝑃) → ∃𝑎 ∈ 𝑟 (𝐻‘𝑟) = (𝐹‘𝑎))
6	4, 5	anim12dan 617	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) → (∃𝑏 ∈ 𝑠 (𝐻‘𝑠) = (𝐹‘𝑏) ∧ ∃𝑎 ∈ 𝑟 (𝐻‘𝑟) = (𝐹‘𝑎)))
7		eqeq12 2747	. . . . . . . . . . . 12 ⊢ (((𝐻‘𝑠) = (𝐹‘𝑏) ∧ (𝐻‘𝑟) = (𝐹‘𝑎)) → ((𝐻‘𝑠) = (𝐻‘𝑟) ↔ (𝐹‘𝑏) = (𝐹‘𝑎)))
8	7	ancoms 457	. . . . . . . . . . 11 ⊢ (((𝐻‘𝑟) = (𝐹‘𝑎) ∧ (𝐻‘𝑠) = (𝐹‘𝑏)) → ((𝐻‘𝑠) = (𝐻‘𝑟) ↔ (𝐹‘𝑏) = (𝐹‘𝑎)))
9	8	adantl 480	. . . . . . . . . 10 ⊢ (((((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) ∧ 𝑎 ∈ 𝑟) ∧ ((𝐻‘𝑟) = (𝐹‘𝑎) ∧ (𝐻‘𝑠) = (𝐹‘𝑏))) → ((𝐻‘𝑠) = (𝐻‘𝑟) ↔ (𝐹‘𝑏) = (𝐹‘𝑎)))
10		simplll 771	. . . . . . . . . . . 12 ⊢ ((((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) ∧ 𝑎 ∈ 𝑟) → 𝐹 Fn 𝐴)
11		simpllr 772	. . . . . . . . . . . 12 ⊢ ((((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) ∧ 𝑎 ∈ 𝑟) → (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃))
12		simpr 483	. . . . . . . . . . . . 13 ⊢ (((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) → 𝑏 ∈ 𝑠)
13	12	anim1i 613	. . . . . . . . . . . 12 ⊢ ((((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) ∧ 𝑎 ∈ 𝑟) → (𝑏 ∈ 𝑠 ∧ 𝑎 ∈ 𝑟))
14	1	elsetpreimafveq 46365	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑏 ∈ 𝑠 ∧ 𝑎 ∈ 𝑟)) → ((𝐹‘𝑏) = (𝐹‘𝑎) → 𝑠 = 𝑟))
15	10, 11, 13, 14	syl3anc 1369	. . . . . . . . . . 11 ⊢ ((((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) ∧ 𝑎 ∈ 𝑟) → ((𝐹‘𝑏) = (𝐹‘𝑎) → 𝑠 = 𝑟))
16	15	adantr 479	. . . . . . . . . 10 ⊢ (((((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) ∧ 𝑎 ∈ 𝑟) ∧ ((𝐻‘𝑟) = (𝐹‘𝑎) ∧ (𝐻‘𝑠) = (𝐹‘𝑏))) → ((𝐹‘𝑏) = (𝐹‘𝑎) → 𝑠 = 𝑟))
17	9, 16	sylbid 239	. . . . . . . . 9 ⊢ (((((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) ∧ 𝑎 ∈ 𝑟) ∧ ((𝐻‘𝑟) = (𝐹‘𝑎) ∧ (𝐻‘𝑠) = (𝐹‘𝑏))) → ((𝐻‘𝑠) = (𝐻‘𝑟) → 𝑠 = 𝑟))
18	17	exp32 419	. . . . . . . 8 ⊢ ((((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) ∧ 𝑎 ∈ 𝑟) → ((𝐻‘𝑟) = (𝐹‘𝑎) → ((𝐻‘𝑠) = (𝐹‘𝑏) → ((𝐻‘𝑠) = (𝐻‘𝑟) → 𝑠 = 𝑟))))
19	18	rexlimdva 3153	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) → (∃𝑎 ∈ 𝑟 (𝐻‘𝑟) = (𝐹‘𝑎) → ((𝐻‘𝑠) = (𝐹‘𝑏) → ((𝐻‘𝑠) = (𝐻‘𝑟) → 𝑠 = 𝑟))))
20	19	com23 86	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) → ((𝐻‘𝑠) = (𝐹‘𝑏) → (∃𝑎 ∈ 𝑟 (𝐻‘𝑟) = (𝐹‘𝑎) → ((𝐻‘𝑠) = (𝐻‘𝑟) → 𝑠 = 𝑟))))
21	20	rexlimdva 3153	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) → (∃𝑏 ∈ 𝑠 (𝐻‘𝑠) = (𝐹‘𝑏) → (∃𝑎 ∈ 𝑟 (𝐻‘𝑟) = (𝐹‘𝑎) → ((𝐻‘𝑠) = (𝐻‘𝑟) → 𝑠 = 𝑟))))
22	21	impd 409	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) → ((∃𝑏 ∈ 𝑠 (𝐻‘𝑠) = (𝐹‘𝑏) ∧ ∃𝑎 ∈ 𝑟 (𝐻‘𝑟) = (𝐹‘𝑎)) → ((𝐻‘𝑠) = (𝐻‘𝑟) → 𝑠 = 𝑟)))
23	6, 22	mpd 15	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) → ((𝐻‘𝑠) = (𝐻‘𝑟) → 𝑠 = 𝑟))
24	23	ralrimivva 3198	. 2 ⊢ (𝐹 Fn 𝐴 → ∀𝑠 ∈ 𝑃 ∀𝑟 ∈ 𝑃 ((𝐻‘𝑠) = (𝐻‘𝑟) → 𝑠 = 𝑟))
25		dff13 7258	. 2 ⊢ (𝐻:𝑃–1-1→(𝐹 “ 𝐴) ↔ (𝐻:𝑃⟶(𝐹 “ 𝐴) ∧ ∀𝑠 ∈ 𝑃 ∀𝑟 ∈ 𝑃 ((𝐻‘𝑠) = (𝐻‘𝑟) → 𝑠 = 𝑟)))
26	3, 24, 25	sylanbrc 581	1 ⊢ (𝐹 Fn 𝐴 → 𝐻:𝑃–1-1→(𝐹 “ 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  {cab 2707  ∀wral 3059  ∃wrex 3068  {csn 4629  ∪ cuni 4909   ↦ cmpt 5232  ^◡ccnv 5676   “ cima 5680   Fn wfn 6539  ⟶wf 6540  –1-1→wf1 6541  ‘cfv 6544

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fv 6552

This theorem is referenced by:  imasetpreimafvbij  46374