Theorem imasetpreimafvbijlemf1 45586

Description: Lemma for imasetpreimafvbij 45588: 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 45583	. 2 ⊢ (𝐹 Fn 𝐴 → 𝐻:𝑃⟶(𝐹 “ 𝐴))
4	1, 2	imasetpreimafvbijlemfv1 45585	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑠 ∈ 𝑃) → ∃𝑏 ∈ 𝑠 (𝐻‘𝑠) = (𝐹‘𝑏))
5	1, 2	imasetpreimafvbijlemfv1 45585	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑟 ∈ 𝑃) → ∃𝑎 ∈ 𝑟 (𝐻‘𝑟) = (𝐹‘𝑎))
6	4, 5	anim12dan 619	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) → (∃𝑏 ∈ 𝑠 (𝐻‘𝑠) = (𝐹‘𝑏) ∧ ∃𝑎 ∈ 𝑟 (𝐻‘𝑟) = (𝐹‘𝑎)))
7		eqeq12 2753	. . . . . . . . . . . 12 ⊢ (((𝐻‘𝑠) = (𝐹‘𝑏) ∧ (𝐻‘𝑟) = (𝐹‘𝑎)) → ((𝐻‘𝑠) = (𝐻‘𝑟) ↔ (𝐹‘𝑏) = (𝐹‘𝑎)))
8	7	ancoms 459	. . . . . . . . . . 11 ⊢ (((𝐻‘𝑟) = (𝐹‘𝑎) ∧ (𝐻‘𝑠) = (𝐹‘𝑏)) → ((𝐻‘𝑠) = (𝐻‘𝑟) ↔ (𝐹‘𝑏) = (𝐹‘𝑎)))
9	8	adantl 482	. . . . . . . . . 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 773	. . . . . . . . . . . 12 ⊢ ((((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) ∧ 𝑎 ∈ 𝑟) → 𝐹 Fn 𝐴)
11		simpllr 774	. . . . . . . . . . . 12 ⊢ ((((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) ∧ 𝑎 ∈ 𝑟) → (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃))
12		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) → 𝑏 ∈ 𝑠)
13	12	anim1i 615	. . . . . . . . . . . 12 ⊢ ((((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) ∧ 𝑎 ∈ 𝑟) → (𝑏 ∈ 𝑠 ∧ 𝑎 ∈ 𝑟))
14	1	elsetpreimafveq 45579	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑏 ∈ 𝑠 ∧ 𝑎 ∈ 𝑟)) → ((𝐹‘𝑏) = (𝐹‘𝑎) → 𝑠 = 𝑟))
15	10, 11, 13, 14	syl3anc 1371	. . . . . . . . . . 11 ⊢ ((((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) ∧ 𝑎 ∈ 𝑟) → ((𝐹‘𝑏) = (𝐹‘𝑎) → 𝑠 = 𝑟))
16	15	adantr 481	. . . . . . . . . 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 421	. . . . . . . 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 3152	. . . . . . 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 3152	. . . . 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 411	. . . 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 3197	. 2 ⊢ (𝐹 Fn 𝐴 → ∀𝑠 ∈ 𝑃 ∀𝑟 ∈ 𝑃 ((𝐻‘𝑠) = (𝐻‘𝑟) → 𝑠 = 𝑟))
25		dff13 7202	. 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 583	1 ⊢ (𝐹 Fn 𝐴 → 𝐻:𝑃–1-1→(𝐹 “ 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2713  ∀wral 3064  ∃wrex 3073  {csn 4586  ∪ cuni 4865   ↦ cmpt 5188  ^◡ccnv 5632   “ cima 5636   Fn wfn 6491  ⟶wf 6492  –1-1→wf1 6493  ‘cfv 6496

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fv 6504

This theorem is referenced by:  imasetpreimafvbij  45588