Theorem imasetpreimafvbijlemf1 47409

Description: Lemma for imasetpreimafvbij 47411: 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 47406	. 2 ⊢ (𝐹 Fn 𝐴 → 𝐻:𝑃⟶(𝐹 “ 𝐴))
4	1, 2	imasetpreimafvbijlemfv1 47408	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑠 ∈ 𝑃) → ∃𝑏 ∈ 𝑠 (𝐻‘𝑠) = (𝐹‘𝑏))
5	1, 2	imasetpreimafvbijlemfv1 47408	. . . . 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 2747	. . . . . . . . . . . 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 774	. . . . . . . . . . . 12 ⊢ ((((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) ∧ 𝑎 ∈ 𝑟) → 𝐹 Fn 𝐴)
11		simpllr 775	. . . . . . . . . . . 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 615	. . . . . . . . . . . 12 ⊢ ((((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) ∧ 𝑎 ∈ 𝑟) → (𝑏 ∈ 𝑠 ∧ 𝑎 ∈ 𝑟))
14	1	elsetpreimafveq 47402	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑏 ∈ 𝑠 ∧ 𝑎 ∈ 𝑟)) → ((𝐹‘𝑏) = (𝐹‘𝑎) → 𝑠 = 𝑟))
15	10, 11, 13, 14	syl3anc 1373	. . . . . . . . . . 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 240	. . . . . . . . 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 3135	. . . . . . 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 3135	. . . . 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 3181	. 2 ⊢ (𝐹 Fn 𝐴 → ∀𝑠 ∈ 𝑃 ∀𝑟 ∈ 𝑃 ((𝐻‘𝑠) = (𝐻‘𝑟) → 𝑠 = 𝑟))
25		dff13 7232	. 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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2708  ∀wral 3045  ∃wrex 3054  {csn 4592  ∪ cuni 4874   ↦ cmpt 5191  ^◡ccnv 5640   “ cima 5644   Fn wfn 6509  ⟶wf 6510  –1-1→wf1 6511  ‘cfv 6514

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fv 6522

This theorem is referenced by:  imasetpreimafvbij  47411