Theorem imasetpreimafvbijlemf1 44744

Description: Lemma for imasetpreimafvbij 44746: 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 44741	. 2 ⊢ (𝐹 Fn 𝐴 → 𝐻:𝑃⟶(𝐹 “ 𝐴))
4	1, 2	imasetpreimafvbijlemfv1 44743	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑠 ∈ 𝑃) → ∃𝑏 ∈ 𝑠 (𝐻‘𝑠) = (𝐹‘𝑏))
5	1, 2	imasetpreimafvbijlemfv1 44743	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑟 ∈ 𝑃) → ∃𝑎 ∈ 𝑟 (𝐻‘𝑟) = (𝐹‘𝑎))
6	4, 5	anim12dan 618	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) → (∃𝑏 ∈ 𝑠 (𝐻‘𝑠) = (𝐹‘𝑏) ∧ ∃𝑎 ∈ 𝑟 (𝐻‘𝑟) = (𝐹‘𝑎)))
7		eqeq12 2755	. . . . . . . . . . . 12 ⊢ (((𝐻‘𝑠) = (𝐹‘𝑏) ∧ (𝐻‘𝑟) = (𝐹‘𝑎)) → ((𝐻‘𝑠) = (𝐻‘𝑟) ↔ (𝐹‘𝑏) = (𝐹‘𝑎)))
8	7	ancoms 458	. . . . . . . . . . 11 ⊢ (((𝐻‘𝑟) = (𝐹‘𝑎) ∧ (𝐻‘𝑠) = (𝐹‘𝑏)) → ((𝐻‘𝑠) = (𝐻‘𝑟) ↔ (𝐹‘𝑏) = (𝐹‘𝑎)))
9	8	adantl 481	. . . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ (((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) → 𝑏 ∈ 𝑠)
13	12	anim1i 614	. . . . . . . . . . . 12 ⊢ ((((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) ∧ 𝑎 ∈ 𝑟) → (𝑏 ∈ 𝑠 ∧ 𝑎 ∈ 𝑟))
14	1	elsetpreimafveq 44737	. . . . . . . . . . . 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 480	. . . . . . . . . 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 420	. . . . . . . 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 3212	. . . . . . 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 3212	. . . . 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 410	. . . 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 3114	. 2 ⊢ (𝐹 Fn 𝐴 → ∀𝑠 ∈ 𝑃 ∀𝑟 ∈ 𝑃 ((𝐻‘𝑠) = (𝐻‘𝑟) → 𝑠 = 𝑟))
25		dff13 7109	. 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 582	1 ⊢ (𝐹 Fn 𝐴 → 𝐻:𝑃–1-1→(𝐹 “ 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  ∃wrex 3064  {csn 4558  ∪ cuni 4836   ↦ cmpt 5153  ^◡ccnv 5579   “ cima 5583   Fn wfn 6413  ⟶wf 6414  –1-1→wf1 6415  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fv 6426

This theorem is referenced by:  imasetpreimafvbij  44746